Theorem 1enumen 35428

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 10151 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 35429 for a version that uses the card function, 1enumkard 35518 for a version that uses the kard function , and 1enum 35542 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 9051	. . 3 ⊢ (𝐴 ∈ V → (𝐴 × 1o) ≈ 𝐴)
2	1	ensymd 9002	. 2 ⊢ (𝐴 ∈ V → 𝐴 ≈ (𝐴 × 1o))
3		iunid 5027	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
4	3	xpeq1i 5688	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 1o) = (𝐴 × 1o)
5		xpiundir 5734	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 1o) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 1o)
6	4, 5	eqtr3i 2794	. 2 ⊢ (𝐴 × 1o) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 1o)
7	2, 6	breqtrdi 5154	1 ⊢ (𝐴 ∈ V → 𝐴 ≈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 1o))

Syntax hints:   → wi 4   ∈ wcel 2149  Vcvv 3461  {csn 4592  ∪ ciun 4958   class class class wbr 5111   × cxp 5660  1_oc1o 8446   ≈ cen 8940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-suc 6367  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-1o 8453  df-er 8694  df-en 8944

This theorem is referenced by:  1enumcard  35429  1enumkard  35518