Theorem iinon 2948

Description: The indexed intersection of a non-empty class of ordinal numbers is an ordinal number. B normally has free variable x as a parameter. Note that A may be a proper class.

Hypothesis

Ref	Expression
iinon.1	⊢ B ∈ V

Assertion

Ref	Expression
iinon	⊢ ((∀x ∈ A B ∈ On ∧ ¬ A = ∅) → ∩x ∈ A B ∈ On)

Distinct variable group(s):   x,A

Proof of Theorem iinon

Step	Hyp	Ref	Expression
1		oninton 2267	. . . 4 ⊢ (({y∣∃x ∈ A y = B} ⊆ On ∧ ¬ {y∣∃x ∈ A y = B} = ∅) → ∩{y∣∃x ∈ A y = B} ∈ On)
2		19.42v 966	. . . . . . . . 9 ⊢ (∃y(x ∈ A ∧ y = B) ↔ (x ∈ A ∧ ∃y y = B))
3		iinon.1	. . . . . . . . . 10 ⊢ B ∈ V
4	3	isseti 1352	. . . . . . . . 9 ⊢ ∃y y = B
5	2, 4	mpbiranr 548	. . . . . . . 8 ⊢ (∃y(x ∈ A ∧ y = B) ↔ x ∈ A)
6	5	biex 733	. . . . . . 7 ⊢ (∃x∃y(x ∈ A ∧ y = B) ↔ ∃x x ∈ A)
7		excom 728	. . . . . . 7 ⊢ (∃x∃y(x ∈ A ∧ y = B) ↔ ∃y∃x(x ∈ A ∧ y = B))
8	6, 7	bitr3 153	. . . . . 6 ⊢ (∃x x ∈ A ↔ ∃y∃x(x ∈ A ∧ y = B))
9		df-rex 1206	. . . . . . 7 ⊢ (∃x ∈ A y = B ↔ ∃x(x ∈ A ∧ y = B))
10	9	biex 733	. . . . . 6 ⊢ (∃y∃x ∈ A y = B ↔ ∃y∃x(x ∈ A ∧ y = B))
11	8, 10	bitr4 154	. . . . 5 ⊢ (∃x x ∈ A ↔ ∃y∃x ∈ A y = B)
12		n0 1714	. . . . 5 ⊢ (¬ A = ∅ ↔ ∃x x ∈ A)
13		abn0 1715	. . . . 5 ⊢ (¬ {y∣∃x ∈ A y = B} = ∅ ↔ ∃y∃x ∈ A y = B)
14	11, 12, 13	3bitr4 158	. . . 4 ⊢ (¬ A = ∅ ↔ ¬ {y∣∃x ∈ A y = B} = ∅)
15	1, 14	sylan2b 347	. . 3 ⊢ (({y∣∃x ∈ A y = B} ⊆ On ∧ ¬ A = ∅) → ∩{y∣∃x ∈ A y = B} ∈ On)
16		hbra1 1237	. . . . . . 7 ⊢ (∀x ∈ A B ∈ On → ∀x∀x ∈ A B ∈ On)
17		ax-17 925	. . . . . . 7 ⊢ (y ∈ On → ∀x y ∈ On)
18		ra4 1243	. . . . . . . 8 ⊢ (∀x ∈ A B ∈ On → (x ∈ A → B ∈ On))
19		eleq1a 1158	. . . . . . . 8 ⊢ (B ∈ On → (y = B → y ∈ On))
20	18, 19	syl6 23	. . . . . . 7 ⊢ (∀x ∈ A B ∈ On → (x ∈ A → (y = B → y ∈ On)))
21	16, 17, 20	r19.23ad 1285	. . . . . 6 ⊢ (∀x ∈ A B ∈ On → (∃x ∈ A y = B → y ∈ On))
22		abid 1094	. . . . . 6 ⊢ (y ∈ {y∣∃x ∈ A y = B} ↔ ∃x ∈ A y = B)
23	21, 22	syl5ib 181	. . . . 5 ⊢ (∀x ∈ A B ∈ On → (y ∈ {y∣∃x ∈ A y = B} → y ∈ On))
24	23	19.21aiv 943	. . . 4 ⊢ (∀x ∈ A B ∈ On → ∀y(y ∈ {y∣∃x ∈ A y = B} → y ∈ On))
25		hbab1 1095	. . . . 5 ⊢ (z ∈ {y∣∃x ∈ A y = B} → ∀y z ∈ {y∣∃x ∈ A y = B})
26		ax-17 925	. . . . 5 ⊢ (z ∈ On → ∀y z ∈ On)
27	25, 26	dfss2f 1499	. . . 4 ⊢ ({y∣∃x ∈ A y = B} ⊆ On ↔ ∀y(y ∈ {y∣∃x ∈ A y = B} → y ∈ On))
28	24, 27	sylibr 175	. . 3 ⊢ (∀x ∈ A B ∈ On → {y∣∃x ∈ A y = B} ⊆ On)
29	15, 28	sylan 343	. 2 ⊢ ((∀x ∈ A B ∈ On ∧ ¬ A = ∅) → ∩{y∣∃x ∈ A y = B} ∈ On)
30	3	dfiin2 2015	. . 3 ⊢ ∩x ∈ A B = ∩{y∣∃x ∈ A y = B}
31	30	eleq1i 1152	. 2 ⊢ (∩x ∈ A B ∈ On ↔ ∩{y∣∃x ∈ A y = B} ∈ On)
32	29, 31	sylibr 175	1 ⊢ ((∀x ∈ A B ∈ On ∧ ¬ A = ∅) → ∩x ∈ A B ∈ On)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  Vcvv 1348   ⊆ wss 1487  ∅c0 1707  ∩cint 1965  ∩ciin 1995  Oncon0 2199

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-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-iin 1997  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203

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