Theorem 1enumcard 35724

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 does not depend on AC, but it is only meaningful for numerable sets. See 1enumen 35723 and 1enumkard 35840 for versions that are meaningful for non-numerable sets, and see 1enum 35904 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 35723	. 2 ⊢ (𝐴 ∈ V → 𝐴 ≈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 1o))
2		carden2b 10048	. 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 2144  Vcvv 3450  {csn 4583  ∪ ciun 4950   class class class wbr 5102   × cxp 5648  ‘cfv 6538  1_oc1o 8469   ≈ cen 8970  cardccrd 10016

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-3or 1104  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-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-pss 4014  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-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  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-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-1o 8476  df-er 8717  df-en 8974  df-card 10020

This theorem is used by: (None)