Theorem 1enumen

Description: The Fundamental Theorem of Enumeration (see https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/enu.pdf ), extended to all sets.

The expression U_ x e. A ( { x } X. B ) can be thought of as expressing an indexed disjoint union |_| x e. A B where each B has its elements tagged with the set x that generated it. See the comment directly before undjudom for context on disjoint union as a representation of cardinal addition.

This theorem is not limited to numerable sets, but it also does not depend on AC. See 1enumcard for a version that uses the cardinality function, and see 1enum for a version that uses an explicit sum of complex number 1s.

(Contributed by BTernaryTau, 26-Jun-2026)

	Ref	Expression
Assertion	1enumen	|- ( A e. _V -> A ~~ U_ x e. A ( { x } X. 1o ) )

Proof

Step	Hyp	Ref	Expression
1		xp1en	|- ( A e. _V -> ( A X. 1o ) ~~ A )
2	1	ensymd	|- ( A e. _V -> A ~~ ( A X. 1o ) )
3		iunid	|- U_ x e. A { x } = A
4	3	xpeq1i	|- ( U_ x e. A { x } X. 1o ) = ( A X. 1o )
5		xpiundir	|- ( U_ x e. A { x } X. 1o ) = U_ x e. A ( { x } X. 1o )
6	4 5	eqtr3i	|- ( A X. 1o ) = U_ x e. A ( { x } X. 1o )
7	2 6	breqtrdi	|- ( A e. _V -> A ~~ U_ x e. A ( { x } X. 1o ) )