Theorem cdaval 8875

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 9252, carddom 9255, and cardsdom 9256. 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	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))

Proof of Theorem cdaval

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3185	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		elex 3185	. 2 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
3		p0ex 4779	. . . . . 6 ⊢ {∅} ∈ V
4		xpexg 6858	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ {∅} ∈ V) → (𝐴 × {∅}) ∈ V)
5	3, 4	mpan2 703	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 × {∅}) ∈ V)
6		snex 4835	. . . . . 6 ⊢ {1𝑜} ∈ V
7		xpexg 6858	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ {1𝑜} ∈ V) → (𝐵 × {1𝑜}) ∈ V)
8	6, 7	mpan2 703	. . . . 5 ⊢ (𝐵 ∈ V → (𝐵 × {1𝑜}) ∈ V)
9	5, 8	anim12i 588	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴 × {∅}) ∈ V ∧ (𝐵 × {1𝑜}) ∈ V))
10		unexb 6856	. . . 4 ⊢ (((𝐴 × {∅}) ∈ V ∧ (𝐵 × {1𝑜}) ∈ V) ↔ ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})) ∈ V)
11	9, 10	sylib 207	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})) ∈ V)
12		xpeq1 5052	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 × {∅}) = (𝐴 × {∅}))
13	12	uneq1d 3728	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 × {∅}) ∪ (𝑦 × {1𝑜})) = ((𝐴 × {∅}) ∪ (𝑦 × {1𝑜})))
14		xpeq1 5052	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 × {1𝑜}) = (𝐵 × {1𝑜}))
15	14	uneq2d 3729	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 × {∅}) ∪ (𝑦 × {1𝑜})) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))
16		df-cda 8873	. . . 4 ⊢ +𝑐 = (𝑥 ∈ V, 𝑦 ∈ V ↦ ((𝑥 × {∅}) ∪ (𝑦 × {1𝑜})))
17	13, 15, 16	ovmpt2g 6693	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})) ∈ V) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))
18	11, 17	mpd3an3 1417	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))
19	1, 2, 18	syl2an 493	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538  ∅c0 3874  {csn 4125   × cxp 5036  (class class class)co 6549  1_𝑜c1o 7440   +_𝑐 ccda 8872

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 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

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 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-cda 8873

This theorem is referenced by:  uncdadom  8876  cdaun  8877  cdaen  8878  cda1dif  8881  pm110.643  8882  xp2cda  8885  cdacomen  8886  cdaassen  8887  xpcdaen  8888  mapcdaen  8889  cdadom1  8891  cdaxpdom  8894  cdafi  8895  cdainf  8897  infcda1  8898  pwcdadom  8921  isfin4-3  9020  alephadd  9278  canthp1lem2  9354  xpsc  16040