Theorem 1enumen 35640

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

The expression ∪ 𝑥 ∈ 𝐴({𝑥} × 𝐵) can be thought of as expressing an indexed disjoint union ⊔ 𝑥 ∈ 𝐴𝐵 where each 𝐵 has its elements tagged with the set 𝑥 that generated it. See the comment directly before undjudom 10203 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 35641 for a version that uses the card function, 1enumkard 35759 for a version that uses the kard function , and 1enum 35819 for a version that uses an explicit sum of complex number 1s.

(Contributed by BTernaryTau, 26-Jun-2026.)

Assertion

Ref	Expression
1enumen	⊢ (𝐴 ∈ V → 𝐴 ≈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 1o))

Distinct variable group:   𝑥,𝐴

Proof of Theorem 1enumen

Step	Hyp	Ref	Expression
1		xp1en 9061	. . 3 ⊢ (𝐴 ∈ V → (𝐴 × 1o) ≈ 𝐴)
2	1	ensymd 9011	. 2 ⊢ (𝐴 ∈ V → 𝐴 ≈ (𝐴 × 1o))
3		iunid 5019	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
4	3	xpeq1i 5674	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 1o) = (𝐴 × 1o)
5		xpiundir 5720	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 1o) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 1o)
6	4, 5	eqtr3i 2785	. 2 ⊢ (𝐴 × 1o) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 1o)
7	2, 6	breqtrdi 5146	1 ⊢ (𝐴 ∈ V → 𝐴 ≈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 1o))

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Vcvv 3450  {csn 4584  ∪ ciun 4951   class class class wbr 5103   × cxp 5646  1_oc1o 8448   ≈ cen 8949

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-suc 6358  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-1o 8455  df-er 8696  df-en 8953

This theorem is used by:  1enumcard  35641  1enumkard  35759