CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passverified 2026-08-06 (claude-opus-5)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

If $x \cup {x} = y \cup {y}$ then $x = y$

For all sets $x$ and $y$ , if $x \cup {x} = y \cup {y}$ then $x = y$ .

Given: sets $x$ and $y$ with $x \cup {x} = y \cup {y}$ ; the Axiom of Foundation, through the result cited as [L3].

[L1]

[L2]

${x} : = {x, x}$ , the singleton of $x$ , is the set whose only element is $x$ (The unordered pair ${x, y}$ and the singleton ${x} = {x, x}$ ).

[L3]

[L4]

technique · direct

1.1L1L2L4

$x \in {x}$ , so $x \in x \cup {x}$ ; likewise $y \in y \cup {y}$ .

2.1L1L2givenstep 1.1

By hypothesis the two sets are equal, so $x \in y \cup {y}$ and $y \in x \cup {x}$ ; hence $x \in y$ or $x = y$ , and $y \in x$ or $y = x$ .

3.1L3step 2.1∎

If $x \neq y$ then both alternatives $x = y$ and $y = x$ fail, leaving $x \in y$ and $y \in x$ , which is impossible; therefore $x = y$ .

The general statement, and its special case for $ω$ . This holds for all sets and is proved from Foundation. The corresponding clause for the natural numbers, that the successor is injective on $ω$ , is part (P2) of The von Neumann naturals form a Peano system ↗, which proves it by a different route that does not use Foundation.

Depends on