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	$$⊢A\in \mathrm{V}\to A\approx \bigcup _{x\in A}\left(\left\{x\right\}\times 1_{𝑜}\right)$$

Proof

Step	Hyp	Ref	Expression
1		xp1en	$$⊢A\in \mathrm{V}\to \left(A\times 1_{𝑜}\right)\approx A$$
2	1	ensymd	$$⊢A\in \mathrm{V}\to A\approx \left(A\times 1_{𝑜}\right)$$
3		iunid	$$⊢\bigcup _{x\in A}\left\{x\right\}=A$$
4	3	xpeq1i	$$⊢\bigcup _{x\in A}\left\{x\right\}\times 1_{𝑜}=A\times 1_{𝑜}$$
5		xpiundir	$$⊢\bigcup _{x\in A}\left\{x\right\}\times 1_{𝑜}=\bigcup _{x\in A}\left(\left\{x\right\}\times 1_{𝑜}\right)$$
6	4 5	eqtr3i	$$⊢A\times 1_{𝑜}=\bigcup _{x\in A}\left(\left\{x\right\}\times 1_{𝑜}\right)$$
7	2 6	breqtrdi	$$⊢A\in \mathrm{V}\to A\approx \bigcup _{x\in A}\left(\left\{x\right\}\times 1_{𝑜}\right)$$