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Theorem cdaval 9030
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 9411, carddom 9414, and cardsdom 9415. 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 3243 . 2 (𝐴𝑉𝐴 ∈ V)
2 elex 3243 . 2 (𝐵𝑊𝐵 ∈ V)
3 p0ex 4883 . . . . . 6 {∅} ∈ V
4 xpexg 7002 . . . . . 6 ((𝐴 ∈ V ∧ {∅} ∈ V) → (𝐴 × {∅}) ∈ V)
53, 4mpan2 707 . . . . 5 (𝐴 ∈ V → (𝐴 × {∅}) ∈ V)
6 snex 4938 . . . . . 6 {1𝑜} ∈ V
7 xpexg 7002 . . . . . 6 ((𝐵 ∈ V ∧ {1𝑜} ∈ V) → (𝐵 × {1𝑜}) ∈ V)
86, 7mpan2 707 . . . . 5 (𝐵 ∈ V → (𝐵 × {1𝑜}) ∈ V)
95, 8anim12i 589 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴 × {∅}) ∈ V ∧ (𝐵 × {1𝑜}) ∈ V))
10 unexb 7000 . . . 4 (((𝐴 × {∅}) ∈ V ∧ (𝐵 × {1𝑜}) ∈ V) ↔ ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})) ∈ V)
119, 10sylib 208 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})) ∈ V)
12 xpeq1 5157 . . . . 5 (𝑥 = 𝐴 → (𝑥 × {∅}) = (𝐴 × {∅}))
1312uneq1d 3799 . . . 4 (𝑥 = 𝐴 → ((𝑥 × {∅}) ∪ (𝑦 × {1𝑜})) = ((𝐴 × {∅}) ∪ (𝑦 × {1𝑜})))
14 xpeq1 5157 . . . . 5 (𝑦 = 𝐵 → (𝑦 × {1𝑜}) = (𝐵 × {1𝑜}))
1514uneq2d 3800 . . . 4 (𝑦 = 𝐵 → ((𝐴 × {∅}) ∪ (𝑦 × {1𝑜})) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))
16 df-cda 9028 . . . 4 +𝑐 = (𝑥 ∈ V, 𝑦 ∈ V ↦ ((𝑥 × {∅}) ∪ (𝑦 × {1𝑜})))
1713, 15, 16ovmpt2g 6837 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})) ∈ V) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))
1811, 17mpd3an3 1465 . 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 1523  wcel 2030  Vcvv 3231  cun 3605  c0 3948  {csn 4210   × cxp 5141  (class class class)co 6690  1𝑜c1o 7598   +𝑐 ccda 9027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-cda 9028
This theorem is referenced by:  uncdadom  9031  cdaun  9032  cdaen  9033  cda1dif  9036  pm110.643  9037  xp2cda  9040  cdacomen  9041  cdaassen  9042  xpcdaen  9043  mapcdaen  9044  cdadom1  9046  cdaxpdom  9049  cdafi  9050  cdainf  9052  infcda1  9053  pwcdadom  9076  isfin4-3  9175  alephadd  9437  canthp1lem2  9513  xpsc  16264
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