Theorem 1enumcard 35495

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 10158 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 35494 and 1enumkard 35593 for versions that are meaningful for non-numerable sets, and see 1enum 35617 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 35494	. 2 ⊢ (𝐴 ∈ V → 𝐴 ≈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 1o))
2		carden2b 9960	. 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 1569   ∈ wcel 2142  Vcvv 3454  {csn 4588  ∪ ciun 4955   class class class wbr 5108   × cxp 5658  ‘cfv 6536  1_oc1o 8444   ≈ cen 8938  cardccrd 9928

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  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 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-ord 6363  df-on 6364  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-1o 8451  df-er 8692  df-en 8942  df-card 9932

This theorem is used by: (None)