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Theorem cdaval 8853
Description: Value of cardinal addition. Definition of cardinal sum in [Mendelson] p. 258. For cardinal arithmetic, we follow Mendelson. Rather than defining operations restricted to cardinal numbers, we use this disjoint union operation for addition, while Cartesian product and set exponentiation stand in for cardinal multiplication and exponentiation. Equinumerosity and dominance serve the roles of equality and ordering. If we wanted to, we could easily convert our theorems to actual cardinal number operations via carden 9230, carddom 9233, and cardsdom 9234. The advantage of Mendelson's approach is that we can directly use many equinumerosity theorems that we already have available. (Contributed by NM, 24-Sep-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
Assertion
Ref Expression
cdaval ((𝐴𝑉𝐵𝑊) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))

Proof of Theorem cdaval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3185 . 2 (𝐴𝑉𝐴 ∈ V)
2 elex 3185 . 2 (𝐵𝑊𝐵 ∈ V)
3 p0ex 4774 . . . . . 6 {∅} ∈ V
4 xpexg 6836 . . . . . 6 ((𝐴 ∈ V ∧ {∅} ∈ V) → (𝐴 × {∅}) ∈ V)
53, 4mpan2 703 . . . . 5 (𝐴 ∈ V → (𝐴 × {∅}) ∈ V)
6 snex 4830 . . . . . 6 {1𝑜} ∈ V
7 xpexg 6836 . . . . . 6 ((𝐵 ∈ V ∧ {1𝑜} ∈ V) → (𝐵 × {1𝑜}) ∈ V)
86, 7mpan2 703 . . . . 5 (𝐵 ∈ V → (𝐵 × {1𝑜}) ∈ V)
95, 8anim12i 588 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴 × {∅}) ∈ V ∧ (𝐵 × {1𝑜}) ∈ V))
10 unexb 6834 . . . 4 (((𝐴 × {∅}) ∈ V ∧ (𝐵 × {1𝑜}) ∈ V) ↔ ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})) ∈ V)
119, 10sylib 207 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})) ∈ V)
12 xpeq1 5042 . . . . 5 (𝑥 = 𝐴 → (𝑥 × {∅}) = (𝐴 × {∅}))
1312uneq1d 3728 . . . 4 (𝑥 = 𝐴 → ((𝑥 × {∅}) ∪ (𝑦 × {1𝑜})) = ((𝐴 × {∅}) ∪ (𝑦 × {1𝑜})))
14 xpeq1 5042 . . . . 5 (𝑦 = 𝐵 → (𝑦 × {1𝑜}) = (𝐵 × {1𝑜}))
1514uneq2d 3729 . . . 4 (𝑦 = 𝐵 → ((𝐴 × {∅}) ∪ (𝑦 × {1𝑜})) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))
16 df-cda 8851 . . . 4 +𝑐 = (𝑥 ∈ V, 𝑦 ∈ V ↦ ((𝑥 × {∅}) ∪ (𝑦 × {1𝑜})))
1713, 15, 16ovmpt2g 6671 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})) ∈ V) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))
1811, 17mpd3an3 1417 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))
191, 2, 18syl2an 493 1 ((𝐴𝑉𝐵𝑊) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  cun 3538  c0 3874  {csn 4125   × cxp 5026  (class class class)co 6527  1𝑜c1o 7418   +𝑐 ccda 8850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-br 4579  df-opab 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-cda 8851
This theorem is referenced by:  uncdadom  8854  cdaun  8855  cdaen  8856  cda1dif  8859  pm110.643  8860  xp2cda  8863  cdacomen  8864  cdaassen  8865  xpcdaen  8866  mapcdaen  8867  cdadom1  8869  cdaxpdom  8872  cdafi  8873  cdainf  8875  infcda1  8876  pwcdadom  8899  isfin4-3  8998  alephadd  9256  canthp1lem2  9332  xpsc  15989
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