Theorem cdacomen 3724

Description: Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258.

Hypotheses

Ref	Expression
cdacomen.1	⊢ A ∈ V
cdacomen.2	⊢ B ∈ V

Assertion

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

Proof of Theorem cdacomen

Step	Hyp	Ref	Expression
1		0ne1oOLD 3113	. . . . 5 ⊢ ¬ ∅ = 1o
2		xpsndisj 2655	. . . . 5 ⊢ (¬ ∅ = 1o → ((A × {∅}) ∩ (B × {1o})) = ∅)
3	1, 2	ax-mp 6	. . . 4 ⊢ ((A × {∅}) ∩ (B × {1o})) = ∅
4		cleqcom 1103	. . . . . 6 ⊢ (∅ = 1o ↔ 1o = ∅)
5	1, 4	mtbi 166	. . . . 5 ⊢ ¬ 1o = ∅
6		xpsndisj 2655	. . . . 5 ⊢ (¬ 1o = ∅ → ((A × {1o}) ∩ (B × {∅})) = ∅)
7	5, 6	ax-mp 6	. . . 4 ⊢ ((A × {1o}) ∩ (B × {∅})) = ∅
8		cdacomen.1	. . . . . . 7 ⊢ A ∈ V
9		0ex 1745	. . . . . . . 8 ⊢ ∅ ∈ V
10	8, 9	xpsnen 3339	. . . . . . 7 ⊢ (A × {∅}) ≈ A
11		1o 3109	. . . . . . . . 9 ⊢ 1o ∈ On
12	11	elisseti 1355	. . . . . . . 8 ⊢ 1o ∈ V
13	8, 12	xpsnen 3339	. . . . . . 7 ⊢ (A × {1o}) ≈ A
14	8, 10, 13	entr4 3324	. . . . . 6 ⊢ (A × {∅}) ≈ (A × {1o})
15		cdacomen.2	. . . . . . 7 ⊢ B ∈ V
16	15, 12	xpsnen 3339	. . . . . . 7 ⊢ (B × {1o}) ≈ B
17	15, 9	xpsnen 3339	. . . . . . 7 ⊢ (B × {∅}) ≈ B
18	15, 16, 17	entr4 3324	. . . . . 6 ⊢ (B × {1o}) ≈ (B × {∅})
19	14, 18	pm3.2i 234	. . . . 5 ⊢ ((A × {∅}) ≈ (A × {1o}) ∧ (B × {1o}) ≈ (B × {∅}))
20		unen 3338	. . . . 5 ⊢ ((((A × {∅}) ≈ (A × {1o}) ∧ (B × {1o}) ≈ (B × {∅})) ∧ (((A × {∅}) ∩ (B × {1o})) = ∅ ∧ ((A × {1o}) ∩ (B × {∅})) = ∅)) → ((A × {∅}) ∪ (B × {1o})) ≈ ((A × {1o}) ∪ (B × {∅})))
21	19, 20	mpan 518	. . . 4 ⊢ ((((A × {∅}) ∩ (B × {1o})) = ∅ ∧ ((A × {1o}) ∩ (B × {∅})) = ∅) → ((A × {∅}) ∪ (B × {1o})) ≈ ((A × {1o}) ∪ (B × {∅})))
22	3, 7, 21	mp2an 520	. . 3 ⊢ ((A × {∅}) ∪ (B × {1o})) ≈ ((A × {1o}) ∪ (B × {∅}))
23		uncom 1604	. . 3 ⊢ ((A × {1o}) ∪ (B × {∅})) = ((B × {∅}) ∪ (A × {1o}))
24	22, 23	breqtr 2080	. 2 ⊢ ((A × {∅}) ∪ (B × {1o})) ≈ ((B × {∅}) ∪ (A × {1o}))
25	8, 15	cdaval 3717	. 2 ⊢ (A +c B) = ((A × {∅}) ∪ (B × {1o}))
26	15, 8	cdaval 3717	. 2 ⊢ (B +c A) = ((B × {∅}) ∪ (A × {1o}))
27	24, 25, 26	3brtr4 2085	1 ⊢ (A +c B) ≈ (B +c A)

Syntax hints:  ¬ wn 1   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∪ cun 1485   ∩ cin 1486  ∅c0 1707  {csn 1808   class class class wbr 2054  Oncon0 2199   × cxp 2408  (class class class)co 3001  1_oc1o 3099   ≈ cen 3271   +_c ccda 3714

This theorem is referenced by:  cdadom2 3728

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-opr 3003  df-oprab 3004  df-1o 3104  df-er 3200  df-en 3274  df-cda 3715

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