Theorem cdaval 9105

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 9486, carddom 9489, and cardsdom 9490. 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 3316	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		elex 3316	. 2 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
3		p0ex 4958	. . . . . 6 ⊢ {∅} ∈ V
4		xpexg 7077	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ {∅} ∈ V) → (𝐴 × {∅}) ∈ V)
5	3, 4	mpan2 709	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 × {∅}) ∈ V)
6		snex 5013	. . . . . 6 ⊢ {1𝑜} ∈ V
7		xpexg 7077	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ {1𝑜} ∈ V) → (𝐵 × {1𝑜}) ∈ V)
8	6, 7	mpan2 709	. . . . 5 ⊢ (𝐵 ∈ V → (𝐵 × {1𝑜}) ∈ V)
9	5, 8	anim12i 591	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴 × {∅}) ∈ V ∧ (𝐵 × {1𝑜}) ∈ V))
10		unexb 7075	. . . 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 5232	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 × {∅}) = (𝐴 × {∅}))
13	12	uneq1d 3874	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 × {∅}) ∪ (𝑦 × {1𝑜})) = ((𝐴 × {∅}) ∪ (𝑦 × {1𝑜})))
14		xpeq1 5232	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 × {1𝑜}) = (𝐵 × {1𝑜}))
15	14	uneq2d 3875	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 × {∅}) ∪ (𝑦 × {1𝑜})) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))
16		df-cda 9103	. . . 4 ⊢ +𝑐 = (𝑥 ∈ V, 𝑦 ∈ V ↦ ((𝑥 × {∅}) ∪ (𝑦 × {1𝑜})))
17	13, 15, 16	ovmpt2g 6912	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})) ∈ V) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))
18	11, 17	mpd3an3 1538	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))
19	1, 2, 18	syl2an 495	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1596   ∈ wcel 2103  Vcvv 3304   ∪ cun 3678  ∅c0 4023  {csn 4285   × cxp 5216  (class class class)co 6765  1_𝑜c1o 7673   +_𝑐 ccda 9102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-iota 5964  df-fun 6003  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-cda 9103

This theorem is referenced by:  uncdadom  9106  cdaun  9107  cdaen  9108  cda1dif  9111  pm110.643  9112  xp2cda  9115  cdacomen  9116  cdaassen  9117  xpcdaen  9118  mapcdaen  9119  cdadom1  9121  cdaxpdom  9124  cdafi  9125  cdainf  9127  infcda1  9128  pwcdadom  9151  isfin4-3  9250  alephadd  9512  canthp1lem2  9588  xpsc  16340