# Subset

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A⊆B means that every element of A is also an element of B.

mko-subset · definition · v0.10.0



## Mathematical statement

For sets A and B in the same universe, A⊆B if and only if, for every object x, x∈A implies x∈B.





## Representative formula (exact TeX)

A\subseteq B





## Assumptions


- A and B are sets over the same ambient type or within the same explicitly specified universe.





## Symbol scope


- A — candidate subset
- B — candidate superset
- ⊆ — non-strict subset relation





## Intuition, definition, and misconceptions

To prove A⊆B, start with an arbitrary x∈A and show that the same x also belongs to B. Listing one shared element is not sufficient.

A⊆B permits A=B, so it is non-strict inclusion. To state that A is a proper subset of B, equality must also be excluded.

The expressions x∈A and A⊆B have different type roles: the first relates an element to a set, while the second relates two sets. In Lean's Set α, the two sets must also share the same element type α.

Common mistakes include treating x∈A and {x}⊆A as the same statement, assuming A⊆B requires A≠B, or checking only some elements before asserting inclusion.





## Adopted derivation guide

[Subset: proving inclusion from an arbitrary element and refuting it by a counterexample](https://ocme.evemisslab.com/en/derivations/derivation-mko-subset-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Set membership](https://ocme.evemisslab.com/en/mko/mko-set-membership/) — The subset relation is defined through an element-wise implication between membership statements.





## Verification boundary

This entry is an AI candidate for the definition of subset. Finite enumeration can recompute a concrete inclusion relation, but it cannot establish universal properties of arbitrary sets.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Bind the exact unfolding of Set.Subset, the Mathlib version, and an axiom receipt while keeping non-strict inclusion, strict inclusion, and set equality distinct.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/subset.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-subset/index.md)