Theorem cdacomen 9318

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 7833	. . . . 5 ⊢ 1o ∈ On
2		xpsneng 8314	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 1o ∈ On) → (𝐴 × {1o}) ≈ 𝐴)
3	1, 2	mpan2 682	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 × {1o}) ≈ 𝐴)
4		0ex 5014	. . . . 5 ⊢ ∅ ∈ V
5		xpsneng 8314	. . . . 5 ⊢ ((𝐵 ∈ V ∧ ∅ ∈ V) → (𝐵 × {∅}) ≈ 𝐵)
6	4, 5	mpan2 682	. . . 4 ⊢ (𝐵 ∈ V → (𝐵 × {∅}) ≈ 𝐵)
7		ensym 8271	. . . . 5 ⊢ ((𝐴 × {1o}) ≈ 𝐴 → 𝐴 ≈ (𝐴 × {1o}))
8		ensym 8271	. . . . 5 ⊢ ((𝐵 × {∅}) ≈ 𝐵 → 𝐵 ≈ (𝐵 × {∅}))
9		incom 4032	. . . . . . 7 ⊢ ((𝐴 × {1o}) ∩ (𝐵 × {∅})) = ((𝐵 × {∅}) ∩ (𝐴 × {1o}))
10		xp01disj 7843	. . . . . . 7 ⊢ ((𝐵 × {∅}) ∩ (𝐴 × {1o})) = ∅
11	9, 10	eqtri 2849	. . . . . 6 ⊢ ((𝐴 × {1o}) ∩ (𝐵 × {∅})) = ∅
12		cdaenun 9311	. . . . . 6 ⊢ ((𝐴 ≈ (𝐴 × {1o}) ∧ 𝐵 ≈ (𝐵 × {∅}) ∧ ((𝐴 × {1o}) ∩ (𝐵 × {∅})) = ∅) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1o}) ∪ (𝐵 × {∅})))
13	11, 12	mp3an3 1578	. . . . 5 ⊢ ((𝐴 ≈ (𝐴 × {1o}) ∧ 𝐵 ≈ (𝐵 × {∅})) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1o}) ∪ (𝐵 × {∅})))
14	7, 8, 13	syl2an 589	. . . 4 ⊢ (((𝐴 × {1o}) ≈ 𝐴 ∧ (𝐵 × {∅}) ≈ 𝐵) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1o}) ∪ (𝐵 × {∅})))
15	3, 6, 14	syl2an 589	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1o}) ∪ (𝐵 × {∅})))
16		cdaval 9307	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 +𝑐 𝐴) = ((𝐵 × {∅}) ∪ (𝐴 × {1o})))
17	16	ancoms 452	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 +𝑐 𝐴) = ((𝐵 × {∅}) ∪ (𝐴 × {1o})))
18		uncom 3984	. . . 4 ⊢ ((𝐵 × {∅}) ∪ (𝐴 × {1o})) = ((𝐴 × {1o}) ∪ (𝐵 × {∅}))
19	17, 18	syl6eq 2877	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 +𝑐 𝐴) = ((𝐴 × {1o}) ∪ (𝐵 × {∅})))
20	15, 19	breqtrrd 4901	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) ≈ (𝐵 +𝑐 𝐴))
21	4	enref 8255	. . . 4 ⊢ ∅ ≈ ∅
22	21	a1i 11	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∅ ≈ ∅)
23		cdafn 9306	. . . . 5 ⊢ +𝑐 Fn (V × V)
24		fndm 6223	. . . . 5 ⊢ ( +𝑐 Fn (V × V) → dom +𝑐 = (V × V))
25	23, 24	ax-mp 5	. . . 4 ⊢ dom +𝑐 = (V × V)
26	25	ndmov 7078	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) = ∅)
27		ancom 454	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V))
28	25	ndmov 7078	. . . 4 ⊢ (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 +𝑐 𝐴) = ∅)
29	27, 28	sylnbi 322	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 +𝑐 𝐴) = ∅)
30	22, 26, 29	3brtr4d 4905	. 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) ≈ (𝐵 +𝑐 𝐴))
31	20, 30	pm2.61i 177	1 ⊢ (𝐴 +𝑐 𝐵) ≈ (𝐵 +𝑐 𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 386   = wceq 1656   ∈ wcel 2164  Vcvv 3414   ∪ cun 3796   ∩ cin 3797  ∅c0 4144  {csn 4397   class class class wbr 4873   × cxp 5340  dom cdm 5342  Oncon0 5963   Fn wfn 6118  (class class class)co 6905  1_oc1o 7819   ≈ cen 8219   +_𝑐 ccda 9304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-ord 5966  df-on 5967  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-1o 7826  df-er 8009  df-en 8223  df-cda 9305

This theorem is referenced by:  cdadom2  9324  cdalepw  9333  infcda  9345  alephadd  9714  gchdomtri  9766  pwxpndom  9803  gchpwdom  9807  gchhar  9816