Theorem cdavalt 3716

Description: Value of cardinal addition. Definition of cardinal sum in [Mendelson] p. 258.

Assertion

Ref	Expression
cdavalt	⊢ ((A ∈ C ∧ B ∈ D) → (A +c B) = ((A × {∅}) ∪ (B × {1o})))

Proof of Theorem cdavalt

Step	Hyp	Ref	Expression
1		p0ex 1885	. . . . . 6 ⊢ {∅} ∈ V
2		xpexg 2489	. . . . . 6 ⊢ ((A ∈ V ∧ {∅} ∈ V) → (A × {∅}) ∈ V)
3	1, 2	mpan2 519	. . . . 5 ⊢ (A ∈ V → (A × {∅}) ∈ V)
4		snex 1859	. . . . . 6 ⊢ {1o} ∈ V
5		xpexg 2489	. . . . . 6 ⊢ ((B ∈ V ∧ {1o} ∈ V) → (B × {1o}) ∈ V)
6	4, 5	mpan2 519	. . . . 5 ⊢ (B ∈ V → (B × {1o}) ∈ V)
7	3, 6	anim12i 268	. . . 4 ⊢ ((A ∈ V ∧ B ∈ V) → ((A × {∅}) ∈ V ∧ (B × {1o}) ∈ V))
8		unexb 1950	. . . 4 ⊢ (((A × {∅}) ∈ V ∧ (B × {1o}) ∈ V) ↔ ((A × {∅}) ∪ (B × {1o})) ∈ V)
9	7, 8	sylib 173	. . 3 ⊢ ((A ∈ V ∧ B ∈ V) → ((A × {∅}) ∪ (B × {1o})) ∈ V)
10		xpeq1 2440	. . . . . 6 ⊢ (x = A → (x × {∅}) = (A × {∅}))
11	10	uneq1d 1610	. . . . 5 ⊢ (x = A → ((x × {∅}) ∪ (y × {1o})) = ((A × {∅}) ∪ (y × {1o})))
12		xpeq1 2440	. . . . . 6 ⊢ (y = B → (y × {1o}) = (B × {1o}))
13	12	uneq2d 1611	. . . . 5 ⊢ (y = B → ((A × {∅}) ∪ (y × {1o})) = ((A × {∅}) ∪ (B × {1o})))
14		df-cda 3715	. . . . . 6 ⊢ +c = {⟨⟨x, y⟩, z⟩∣z = ((x × {∅}) ∪ (y × {1o}))}
15		visset 1350	. . . . . . . . 9 ⊢ x ∈ V
16		visset 1350	. . . . . . . . 9 ⊢ y ∈ V
17	15, 16	pm3.2i 234	. . . . . . . 8 ⊢ (x ∈ V ∧ y ∈ V)
18	17	biantrur 544	. . . . . . 7 ⊢ (z = ((x × {∅}) ∪ (y × {1o})) ↔ ((x ∈ V ∧ y ∈ V) ∧ z = ((x × {∅}) ∪ (y × {1o}))))
19	18	bioprabi 3027	. . . . . 6 ⊢ {⟨⟨x, y⟩, z⟩∣z = ((x × {∅}) ∪ (y × {1o}))} = {⟨⟨x, y⟩, z⟩∣((x ∈ V ∧ y ∈ V) ∧ z = ((x × {∅}) ∪ (y × {1o})))}
20	14, 19	eqtr 1119	. . . . 5 ⊢ +c = {⟨⟨x, y⟩, z⟩∣((x ∈ V ∧ y ∈ V) ∧ z = ((x × {∅}) ∪ (y × {1o})))}
21	11, 13, 20	oprabval2g 3050	. . . 4 ⊢ ((A ∈ V ∧ B ∈ V ∧ ((A × {∅}) ∪ (B × {1o})) ∈ V) → (A +c B) = ((A × {∅}) ∪ (B × {1o})))
22	21	3expa 612	. . 3 ⊢ (((A ∈ V ∧ B ∈ V) ∧ ((A × {∅}) ∪ (B × {1o})) ∈ V) → (A +c B) = ((A × {∅}) ∪ (B × {1o})))
23	9, 22	mpdan 527	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → (A +c B) = ((A × {∅}) ∪ (B × {1o})))
24		elisset 1354	. 2 ⊢ (A ∈ C → A ∈ V)
25		elisset 1354	. 2 ⊢ (B ∈ D → B ∈ V)
26	23, 24, 25	syl2an 349	1 ⊢ ((A ∈ C ∧ B ∈ D) → (A +c B) = ((A × {∅}) ∪ (B × {1o})))

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∪ cun 1485  ∅c0 1707  {csn 1808   × cxp 2408  (class class class)co 3001  {copab2 3002  1_oc1o 3099   +_c ccda 3714

This theorem is referenced by:  cdaval 3717  cdafi 3730

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-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  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-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  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-fv 2438  df-opr 3003  df-oprab 3004  df-cda 3715

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