Theorem 1enumen 35392

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 10126 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 35393 for a version that uses the cardinality function, and see 1enum 35472 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 9037	. . 3 ⊢ (𝐴 ∈ V → (𝐴 × 1o) ≈ 𝐴)
2	1	ensymd 8988	. 2 ⊢ (𝐴 ∈ V → 𝐴 ≈ (𝐴 × 1o))
3		iunid 5020	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
4	3	xpeq1i 5675	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 1o) = (𝐴 × 1o)
5		xpiundir 5721	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 1o) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 1o)
6	4, 5	eqtr3i 2789	. 2 ⊢ (𝐴 × 1o) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 1o)
7	2, 6	breqtrdi 5143	1 ⊢ (𝐴 ∈ V → 𝐴 ≈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 1o))

Syntax hints:   → wi 4   ∈ wcel 2144  Vcvv 3456  {csn 4584  ∪ ciun 4951   class class class wbr 5102   × cxp 5647  1_oc1o 8432   ≈ cen 8926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  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 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-suc 6354  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-1o 8439  df-er 8680  df-en 8930

This theorem is referenced by:  1enumcard  35393