Theorem 1enumen 35586

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 10173 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 35587 for a version that uses the card function, 1enumkard 35685 for a version that uses the kard function , and 1enum 35705 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 9064	. . 3 ⊢ (𝐴 ∈ V → (𝐴 × 1o) ≈ 𝐴)
2	1	ensymd 9014	. 2 ⊢ (𝐴 ∈ V → 𝐴 ≈ (𝐴 × 1o))
3		iunid 5023	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
4	3	xpeq1i 5685	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 1o) = (𝐴 × 1o)
5		xpiundir 5731	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 1o) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 1o)
6	4, 5	eqtr3i 2787	. 2 ⊢ (𝐴 × 1o) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 1o)
7	2, 6	breqtrdi 5150	1 ⊢ (𝐴 ∈ V → 𝐴 ≈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 1o))

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Vcvv 3453  {csn 4587  ∪ ciun 4954   class class class wbr 5107   × cxp 5657  1_oc1o 8451   ≈ cen 8952

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 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-suc 6367  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-1o 8458  df-er 8699  df-en 8956

This theorem is used by:  1enumcard  35587  1enumkard  35685