Subset

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

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• 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.