Theorem cdaval 8992

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 9373, carddom 9376, and cardsdom 9377. 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 3212	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		elex 3212	. 2 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
3		p0ex 4853	. . . . . 6 ⊢ {∅} ∈ V
4		xpexg 6960	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ {∅} ∈ V) → (𝐴 × {∅}) ∈ V)
5	3, 4	mpan2 707	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 × {∅}) ∈ V)
6		snex 4908	. . . . . 6 ⊢ {1𝑜} ∈ V
7		xpexg 6960	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ {1𝑜} ∈ V) → (𝐵 × {1𝑜}) ∈ V)
8	6, 7	mpan2 707	. . . . 5 ⊢ (𝐵 ∈ V → (𝐵 × {1𝑜}) ∈ V)
9	5, 8	anim12i 590	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴 × {∅}) ∈ V ∧ (𝐵 × {1𝑜}) ∈ V))
10		unexb 6958	. . . 4 ⊢ (((𝐴 × {∅}) ∈ V ∧ (𝐵 × {1𝑜}) ∈ V) ↔ ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})) ∈ V)
11	9, 10	sylib 208	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})) ∈ V)
12		xpeq1 5128	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 × {∅}) = (𝐴 × {∅}))
13	12	uneq1d 3766	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 × {∅}) ∪ (𝑦 × {1𝑜})) = ((𝐴 × {∅}) ∪ (𝑦 × {1𝑜})))
14		xpeq1 5128	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 × {1𝑜}) = (𝐵 × {1𝑜}))
15	14	uneq2d 3767	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 × {∅}) ∪ (𝑦 × {1𝑜})) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))
16		df-cda 8990	. . . 4 ⊢ +𝑐 = (𝑥 ∈ V, 𝑦 ∈ V ↦ ((𝑥 × {∅}) ∪ (𝑦 × {1𝑜})))
17	13, 15, 16	ovmpt2g 6795	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})) ∈ V) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))
18	11, 17	mpd3an3 1425	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))
19	1, 2, 18	syl2an 494	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∪ cun 3572  ∅c0 3915  {csn 4177   × cxp 5112  (class class class)co 6650  1_𝑜c1o 7553   +_𝑐 ccda 8989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-cda 8990

This theorem is referenced by:  uncdadom  8993  cdaun  8994  cdaen  8995  cda1dif  8998  pm110.643  8999  xp2cda  9002  cdacomen  9003  cdaassen  9004  xpcdaen  9005  mapcdaen  9006  cdadom1  9008  cdaxpdom  9011  cdafi  9012  cdainf  9014  infcda1  9015  pwcdadom  9038  isfin4-3  9137  alephadd  9399  canthp1lem2  9475  xpsc  16217