Theorem 1enumen 35723

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 10246 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 35724 for a version that uses the card function, 1enumkard 35840 for a version that uses the kard function , and 1enum 35904 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 9082	. . 3 ⊢ (𝐴 ∈ V → (𝐴 × 1o) ≈ 𝐴)
2	1	ensymd 9032	. 2 ⊢ (𝐴 ∈ V → 𝐴 ≈ (𝐴 × 1o))
3		iunid 5018	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
4	3	xpeq1i 5676	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 1o) = (𝐴 × 1o)
5		xpiundir 5722	. . 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 5145	1 ⊢ (𝐴 ∈ V → 𝐴 ≈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 1o))

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  Vcvv 3450  {csn 4583  ∪ ciun 4950   class class class wbr 5102   × cxp 5648  1_oc1o 8469   ≈ cen 8970

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-suc 6368  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-1o 8476  df-er 8717  df-en 8974

This theorem is used by:  1enumcard  35724  1enumkard  35840