Theorem cdacomen 9003

Description: Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 24-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
cdacomen	⊢ (𝐴 +𝑐 𝐵) ≈ (𝐵 +𝑐 𝐴)

Proof of Theorem cdacomen

Step	Hyp	Ref	Expression
1		1on 7567	. . . . 5 ⊢ 1𝑜 ∈ On
2		xpsneng 8045	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 × {1𝑜}) ≈ 𝐴)
3	1, 2	mpan2 707	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 × {1𝑜}) ≈ 𝐴)
4		0ex 4790	. . . . 5 ⊢ ∅ ∈ V
5		xpsneng 8045	. . . . 5 ⊢ ((𝐵 ∈ V ∧ ∅ ∈ V) → (𝐵 × {∅}) ≈ 𝐵)
6	4, 5	mpan2 707	. . . 4 ⊢ (𝐵 ∈ V → (𝐵 × {∅}) ≈ 𝐵)
7		ensym 8005	. . . . 5 ⊢ ((𝐴 × {1𝑜}) ≈ 𝐴 → 𝐴 ≈ (𝐴 × {1𝑜}))
8		ensym 8005	. . . . 5 ⊢ ((𝐵 × {∅}) ≈ 𝐵 → 𝐵 ≈ (𝐵 × {∅}))
9		incom 3805	. . . . . . 7 ⊢ ((𝐴 × {1𝑜}) ∩ (𝐵 × {∅})) = ((𝐵 × {∅}) ∩ (𝐴 × {1𝑜}))
10		xp01disj 7576	. . . . . . 7 ⊢ ((𝐵 × {∅}) ∩ (𝐴 × {1𝑜})) = ∅
11	9, 10	eqtri 2644	. . . . . 6 ⊢ ((𝐴 × {1𝑜}) ∩ (𝐵 × {∅})) = ∅
12		cdaenun 8996	. . . . . 6 ⊢ ((𝐴 ≈ (𝐴 × {1𝑜}) ∧ 𝐵 ≈ (𝐵 × {∅}) ∧ ((𝐴 × {1𝑜}) ∩ (𝐵 × {∅})) = ∅) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅})))
13	11, 12	mp3an3 1413	. . . . 5 ⊢ ((𝐴 ≈ (𝐴 × {1𝑜}) ∧ 𝐵 ≈ (𝐵 × {∅})) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅})))
14	7, 8, 13	syl2an 494	. . . 4 ⊢ (((𝐴 × {1𝑜}) ≈ 𝐴 ∧ (𝐵 × {∅}) ≈ 𝐵) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅})))
15	3, 6, 14	syl2an 494	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅})))
16		cdaval 8992	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 +𝑐 𝐴) = ((𝐵 × {∅}) ∪ (𝐴 × {1𝑜})))
17	16	ancoms 469	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 +𝑐 𝐴) = ((𝐵 × {∅}) ∪ (𝐴 × {1𝑜})))
18		uncom 3757	. . . 4 ⊢ ((𝐵 × {∅}) ∪ (𝐴 × {1𝑜})) = ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅}))
19	17, 18	syl6eq 2672	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 +𝑐 𝐴) = ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅})))
20	15, 19	breqtrrd 4681	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) ≈ (𝐵 +𝑐 𝐴))
21	4	enref 7988	. . . 4 ⊢ ∅ ≈ ∅
22	21	a1i 11	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∅ ≈ ∅)
23		cdafn 8991	. . . . 5 ⊢ +𝑐 Fn (V × V)
24		fndm 5990	. . . . 5 ⊢ ( +𝑐 Fn (V × V) → dom +𝑐 = (V × V))
25	23, 24	ax-mp 5	. . . 4 ⊢ dom +𝑐 = (V × V)
26	25	ndmov 6818	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) = ∅)
27		ancom 466	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V))
28	25	ndmov 6818	. . . 4 ⊢ (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 +𝑐 𝐴) = ∅)
29	27, 28	sylnbi 320	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 +𝑐 𝐴) = ∅)
30	22, 26, 29	3brtr4d 4685	. 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) ≈ (𝐵 +𝑐 𝐴))
31	20, 30	pm2.61i 176	1 ⊢ (𝐴 +𝑐 𝐵) ≈ (𝐵 +𝑐 𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∪ cun 3572   ∩ cin 3573  ∅c0 3915  {csn 4177   class class class wbr 4653   × cxp 5112  dom cdm 5114  Oncon0 5723   Fn wfn 5883  (class class class)co 6650  1_𝑜c1o 7553   ≈ cen 7952   +_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-3or 1038  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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-1o 7560  df-er 7742  df-en 7956  df-cda 8990

This theorem is referenced by:  cdadom2  9009  cdalepw  9018  infcda  9030  alephadd  9399  gchdomtri  9451  pwxpndom  9488  gchpwdom  9492  gchhar  9501