Theorem cdaen 3719

Description: Cardinal addition of equinumerous sets. Exercise 4.56(b) of [Mendelson] p. 258.

Hypotheses

Ref	Expression
cdaen.1	⊢ A ∈ V
cdaen.2	⊢ B ∈ V
cdaen.3	⊢ C ∈ V
cdaen.4	⊢ D ∈ V

Assertion

Ref	Expression
cdaen	⊢ ((A ≈ B ∧ C ≈ D) → (A +c C) ≈ (B +c D))

Proof of Theorem cdaen

Step	Hyp	Ref	Expression
1		0ne1oOLD 3113	. . . . . 6 ⊢ ¬ ∅ = 1o
2		xpsndisj 2655	. . . . . 6 ⊢ (¬ ∅ = 1o → ((A × {∅}) ∩ (C × {1o})) = ∅)
3	1, 2	ax-mp 6	. . . . 5 ⊢ ((A × {∅}) ∩ (C × {1o})) = ∅
4		xpsndisj 2655	. . . . . 6 ⊢ (¬ ∅ = 1o → ((B × {∅}) ∩ (D × {1o})) = ∅)
5	1, 4	ax-mp 6	. . . . 5 ⊢ ((B × {∅}) ∩ (D × {1o})) = ∅
6	3, 5	pm3.2i 234	. . . 4 ⊢ (((A × {∅}) ∩ (C × {1o})) = ∅ ∧ ((B × {∅}) ∩ (D × {1o})) = ∅)
7		unen 3338	. . . 4 ⊢ ((((A × {∅}) ≈ (B × {∅}) ∧ (C × {1o}) ≈ (D × {1o})) ∧ (((A × {∅}) ∩ (C × {1o})) = ∅ ∧ ((B × {∅}) ∩ (D × {1o})) = ∅)) → ((A × {∅}) ∪ (C × {1o})) ≈ ((B × {∅}) ∪ (D × {1o})))
8	6, 7	mpan2 519	. . 3 ⊢ (((A × {∅}) ≈ (B × {∅}) ∧ (C × {1o}) ≈ (D × {1o})) → ((A × {∅}) ∪ (C × {1o})) ≈ ((B × {∅}) ∪ (D × {1o})))
9		cdaen.1	. . . . 5 ⊢ A ∈ V
10		0ex 1745	. . . . . 6 ⊢ ∅ ∈ V
11	9, 10	xpsnen 3339	. . . . 5 ⊢ (A × {∅}) ≈ A
12		enen1 3375	. . . . 5 ⊢ ((A ∈ V ∧ (A × {∅}) ≈ A) → ((A × {∅}) ≈ (B × {∅}) ↔ A ≈ (B × {∅})))
13	9, 11, 12	mp2an 520	. . . 4 ⊢ ((A × {∅}) ≈ (B × {∅}) ↔ A ≈ (B × {∅}))
14		cdaen.2	. . . . 5 ⊢ B ∈ V
15	14, 10	xpsnen 3339	. . . . 5 ⊢ (B × {∅}) ≈ B
16		enen2 3376	. . . . 5 ⊢ ((B ∈ V ∧ (B × {∅}) ≈ B) → (A ≈ (B × {∅}) ↔ A ≈ B))
17	14, 15, 16	mp2an 520	. . . 4 ⊢ (A ≈ (B × {∅}) ↔ A ≈ B)
18	13, 17	bitr 151	. . 3 ⊢ ((A × {∅}) ≈ (B × {∅}) ↔ A ≈ B)
19		cdaen.3	. . . . 5 ⊢ C ∈ V
20		1o 3109	. . . . . . 7 ⊢ 1o ∈ On
21	20	elisseti 1355	. . . . . 6 ⊢ 1o ∈ V
22	19, 21	xpsnen 3339	. . . . 5 ⊢ (C × {1o}) ≈ C
23		enen1 3375	. . . . 5 ⊢ ((C ∈ V ∧ (C × {1o}) ≈ C) → ((C × {1o}) ≈ (D × {1o}) ↔ C ≈ (D × {1o})))
24	19, 22, 23	mp2an 520	. . . 4 ⊢ ((C × {1o}) ≈ (D × {1o}) ↔ C ≈ (D × {1o}))
25		cdaen.4	. . . . 5 ⊢ D ∈ V
26	25, 21	xpsnen 3339	. . . . 5 ⊢ (D × {1o}) ≈ D
27		enen2 3376	. . . . 5 ⊢ ((D ∈ V ∧ (D × {1o}) ≈ D) → (C ≈ (D × {1o}) ↔ C ≈ D))
28	25, 26, 27	mp2an 520	. . . 4 ⊢ (C ≈ (D × {1o}) ↔ C ≈ D)
29	24, 28	bitr 151	. . 3 ⊢ ((C × {1o}) ≈ (D × {1o}) ↔ C ≈ D)
30	8, 18, 29	syl2anbr 351	. 2 ⊢ ((A ≈ B ∧ C ≈ D) → ((A × {∅}) ∪ (C × {1o})) ≈ ((B × {∅}) ∪ (D × {1o})))
31	9, 19	cdaval 3717	. 2 ⊢ (A +c C) = ((A × {∅}) ∪ (C × {1o}))
32	14, 25	cdaval 3717	. 2 ⊢ (B +c D) = ((B × {∅}) ∪ (D × {1o}))
33	30, 31, 32	3brtr4g 2088	1 ⊢ ((A ≈ B ∧ C ≈ D) → (A +c C) ≈ (B +c D))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ 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 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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