Theorem 1enumcard 35587

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 does not depend on AC, but it is only meaningful for numerable sets. See 1enumen 35586 and 1enumkard 35685 for versions that are meaningful for non-numerable sets, and see 1enum 35705 for a version that uses an explicit sum of complex number 1s.

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

Assertion

Ref	Expression
1enumcard	⊢ (𝐴 ∈ V → (card‘𝐴) = (card‘∪ 𝑥 ∈ 𝐴 ({𝑥} × 1o)))

Distinct variable group:   𝑥,𝐴

Proof of Theorem 1enumcard

Step	Hyp	Ref	Expression
1		1enumen 35586	. 2 ⊢ (𝐴 ∈ V → 𝐴 ≈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 1o))
2		carden2b 9975	. 2 ⊢ (𝐴 ≈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 1o) → (card‘𝐴) = (card‘∪ 𝑥 ∈ 𝐴 ({𝑥} × 1o)))
3	1, 2	syl 18	1 ⊢ (𝐴 ∈ V → (card‘𝐴) = (card‘∪ 𝑥 ∈ 𝐴 ({𝑥} × 1o)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3453  {csn 4587  ∪ ciun 4954   class class class wbr 5107   × cxp 5657  ‘cfv 6537  1_oc1o 8451   ≈ cen 8952  cardccrd 9943

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-3or 1104  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-ne 2958  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-pss 3922  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-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  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-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-1o 8458  df-er 8699  df-en 8956  df-card 9947

This theorem is used by: (None)