Theorem cardiun 3665

Description: The indexed union of a set of cardinals is a cardinal.

Assertion

Ref	Expression
cardiun	⊢ (A ∈ C → (∀x ∈ A (card ‘B) = B → (card ‘∪x ∈ A B) = ∪x ∈ A B))

Distinct variable group(s):   x,A

Proof of Theorem cardiun

Step	Hyp	Ref	Expression
1		abrexexg 2913	. . . . . 6 ⊢ (A ∈ C → {z∣∃x ∈ A z = (card ‘B)} ∈ V)
2		visset 1350	. . . . . . . . . 10 ⊢ y ∈ V
3		cleq1 1107	. . . . . . . . . . 11 ⊢ (z = y → (z = (card ‘B) ↔ y = (card ‘B)))
4	3	birexdv 1220	. . . . . . . . . 10 ⊢ (z = y → (∃x ∈ A z = (card ‘B) ↔ ∃x ∈ A y = (card ‘B)))
5	2, 4	elab 1415	. . . . . . . . 9 ⊢ (y ∈ {z∣∃x ∈ A z = (card ‘B)} ↔ ∃x ∈ A y = (card ‘B))
6		fveq2 2832	. . . . . . . . . . . . 13 ⊢ (y = (card ‘B) → (card ‘y) = (card ‘(card ‘B)))
7		cardcard 3655	. . . . . . . . . . . . 13 ⊢ (card ‘(card ‘B)) = (card ‘B)
8	6, 7	syl6eq 1140	. . . . . . . . . . . 12 ⊢ (y = (card ‘B) → (card ‘y) = (card ‘B))
9		id 9	. . . . . . . . . . . 12 ⊢ (y = (card ‘B) → y = (card ‘B))
10	8, 9	eqtr4d 1131	. . . . . . . . . . 11 ⊢ (y = (card ‘B) → (card ‘y) = y)
11	10	a1i 7	. . . . . . . . . 10 ⊢ (x ∈ A → (y = (card ‘B) → (card ‘y) = y))
12	11	r19.23aiv 1284	. . . . . . . . 9 ⊢ (∃x ∈ A y = (card ‘B) → (card ‘y) = y)
13	5, 12	sylbi 174	. . . . . . . 8 ⊢ (y ∈ {z∣∃x ∈ A z = (card ‘B)} → (card ‘y) = y)
14	13	rgen 1247	. . . . . . 7 ⊢ ∀y ∈ {z∣∃x ∈ A z = (card ‘B)} (card ‘y) = y
15		carduni 3664	. . . . . . 7 ⊢ ({z∣∃x ∈ A z = (card ‘B)} ∈ V → (∀y ∈ {z∣∃x ∈ A z = (card ‘B)} (card ‘y) = y → (card ‘∪{z∣∃x ∈ A z = (card ‘B)}) = ∪{z∣∃x ∈ A z = (card ‘B)}))
16	14, 15	mpi 44	. . . . . 6 ⊢ ({z∣∃x ∈ A z = (card ‘B)} ∈ V → (card ‘∪{z∣∃x ∈ A z = (card ‘B)}) = ∪{z∣∃x ∈ A z = (card ‘B)})
17	1, 16	syl 12	. . . . 5 ⊢ (A ∈ C → (card ‘∪{z∣∃x ∈ A z = (card ‘B)}) = ∪{z∣∃x ∈ A z = (card ‘B)})
18		fvex 2838	. . . . . . 7 ⊢ (card ‘B) ∈ V
19	18	dfiun2 2014	. . . . . 6 ⊢ ∪x ∈ A (card ‘B) = ∪{z∣∃x ∈ A z = (card ‘B)}
20	19	fveq2i 2835	. . . . 5 ⊢ (card ‘∪x ∈ A (card ‘B)) = (card ‘∪{z∣∃x ∈ A z = (card ‘B)})
21	17, 20, 19	3eqtr4g 1147	. . . 4 ⊢ (A ∈ C → (card ‘∪x ∈ A (card ‘B)) = ∪x ∈ A (card ‘B))
22	21	adantr 306	. . 3 ⊢ ((A ∈ C ∧ ∀x ∈ A (card ‘B) = B) → (card ‘∪x ∈ A (card ‘B)) = ∪x ∈ A (card ‘B))
23		iuneq2 2006	. . . . 5 ⊢ (∀x ∈ A (card ‘B) = B → ∪x ∈ A (card ‘B) = ∪x ∈ A B)
24	23	adantl 305	. . . 4 ⊢ ((A ∈ C ∧ ∀x ∈ A (card ‘B) = B) → ∪x ∈ A (card ‘B) = ∪x ∈ A B)
25	24	fveq2d 2836	. . 3 ⊢ ((A ∈ C ∧ ∀x ∈ A (card ‘B) = B) → (card ‘∪x ∈ A (card ‘B)) = (card ‘∪x ∈ A B))
26	22, 25, 24	3eqtr3d 1133	. 2 ⊢ ((A ∈ C ∧ ∀x ∈ A (card ‘B) = B) → (card ‘∪x ∈ A B) = ∪x ∈ A B)
27	26	exp 291	1 ⊢ (A ∈ C → (∀x ∈ A (card ‘B) = B → (card ‘∪x ∈ A B) = ∪x ∈ A B))

Syntax hints:   → wi 2   ∧ wa 196   = weq 797  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  Vcvv 1348  ∪cuni 1919  ∪ciun 1994   ‘cfv 2422  cardccrd 3620

This theorem is referenced by:  alephcard 3673

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  ax-reg 1078  ax-ac 1080

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-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  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-iun 1996  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-er 3200  df-en 3274  df-dom 3275  df-sdom 3276  df-card 3623

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