Theorem iunon 2947

Description: The indexed union of a set of ordinal numbers is an ordinal number. B normally has free variable x as a parameter.

Hypotheses

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

Assertion

Ref	Expression
iunon	⊢ (∀x ∈ A B ∈ On → ∪x ∈ A B ∈ On)

Distinct variable group(s):   x,A

Proof of Theorem iunon

Step	Hyp	Ref	Expression
1		hbra1 1237	. . . . . . 7 ⊢ (∀x ∈ A B ∈ On → ∀x∀x ∈ A B ∈ On)
2		ax-17 925	. . . . . . 7 ⊢ (y ∈ On → ∀x y ∈ On)
3		ra4 1243	. . . . . . . 8 ⊢ (∀x ∈ A B ∈ On → (x ∈ A → B ∈ On))
4		eleq1a 1158	. . . . . . . 8 ⊢ (B ∈ On → (y = B → y ∈ On))
5	3, 4	syl6 23	. . . . . . 7 ⊢ (∀x ∈ A B ∈ On → (x ∈ A → (y = B → y ∈ On)))
6	1, 2, 5	r19.23ad 1285	. . . . . 6 ⊢ (∀x ∈ A B ∈ On → (∃x ∈ A y = B → y ∈ On))
7		abid 1094	. . . . . 6 ⊢ (y ∈ {y∣∃x ∈ A y = B} ↔ ∃x ∈ A y = B)
8	6, 7	syl5ib 181	. . . . 5 ⊢ (∀x ∈ A B ∈ On → (y ∈ {y∣∃x ∈ A y = B} → y ∈ On))
9	8	19.21aiv 943	. . . 4 ⊢ (∀x ∈ A B ∈ On → ∀y(y ∈ {y∣∃x ∈ A y = B} → y ∈ On))
10		hbab1 1095	. . . . 5 ⊢ (z ∈ {y∣∃x ∈ A y = B} → ∀y z ∈ {y∣∃x ∈ A y = B})
11		ax-17 925	. . . . 5 ⊢ (z ∈ On → ∀y z ∈ On)
12	10, 11	dfss2f 1499	. . . 4 ⊢ ({y∣∃x ∈ A y = B} ⊆ On ↔ ∀y(y ∈ {y∣∃x ∈ A y = B} → y ∈ On))
13	9, 12	sylibr 175	. . 3 ⊢ (∀x ∈ A B ∈ On → {y∣∃x ∈ A y = B} ⊆ On)
14		iunon.1	. . . . 5 ⊢ A ∈ V
15	14	abrexex 2912	. . . 4 ⊢ {y∣∃x ∈ A y = B} ∈ V
16	15	onuni 2251	. . 3 ⊢ ({y∣∃x ∈ A y = B} ⊆ On → ∪{y∣∃x ∈ A y = B} ∈ On)
17	13, 16	syl 12	. 2 ⊢ (∀x ∈ A B ∈ On → ∪{y∣∃x ∈ A y = B} ∈ On)
18		iunon.2	. . . 4 ⊢ B ∈ V
19	18	dfiun2 2014	. . 3 ⊢ ∪x ∈ A B = ∪{y∣∃x ∈ A y = B}
20	19	eleq1i 1152	. 2 ⊢ (∪x ∈ A B ∈ On ↔ ∪{y∣∃x ∈ A y = B} ∈ On)
21	17, 20	sylibr 175	1 ⊢ (∀x ∈ A B ∈ On → ∪x ∈ A B ∈ On)

Syntax hints:   → wi 2  ∀wal 672  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  Vcvv 1348   ⊆ wss 1487  ∪cuni 1919  ∪ciun 1994  Oncon0 2199

This theorem is referenced by:  oacl 3138  omcl 3139  oecl 3140  rankuni 3533  ranklon 3540  alephon 3671

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-rab 1208  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-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-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

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