Theorem ontopbas 32427

Description: An ordinal number is a topological basis. (Contributed by Chen-Pang He, 8-Oct-2015.)

Assertion

Ref	Expression
ontopbas	|- ( B e. On -> B e. TopBases )

Proof of Theorem ontopbas

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		onelon 5748	. . . . . . . 8 |- ( ( B e. On /\ x e. B ) -> x e. On )
2		onelon 5748	. . . . . . . 8 |- ( ( B e. On /\ y e. B ) -> y e. On )
3	1, 2	anim12dan 882	. . . . . . 7 |- ( ( B e. On /\ ( x e. B /\ y e. B ) ) -> ( x e. On /\ y e. On ) )
4	3	ex 450	. . . . . 6 |- ( B e. On -> ( ( x e. B /\ y e. B ) -> ( x e. On /\ y e. On ) ) )
5		onin 5754	. . . . . 6 |- ( ( x e. On /\ y e. On ) -> ( x i^i y ) e. On )
6	4, 5	syl6 35	. . . . 5 |- ( B e. On -> ( ( x e. B /\ y e. B ) -> ( x i^i y ) e. On ) )
7	6	anc2ri 581	. . . 4 |- ( B e. On -> ( ( x e. B /\ y e. B ) -> ( ( x i^i y ) e. On /\ B e. On ) ) )
8		inss1 3833	. . . . . . 7 |- ( x i^i y ) C_ x
9	8	jctl 564	. . . . . 6 |- ( x e. B -> ( ( x i^i y ) C_ x /\ x e. B ) )
10	9	adantr 481	. . . . 5 |- ( ( x e. B /\ y e. B ) -> ( ( x i^i y ) C_ x /\ x e. B ) )
11	10	a1i 11	. . . 4 |- ( B e. On -> ( ( x e. B /\ y e. B ) -> ( ( x i^i y ) C_ x /\ x e. B ) ) )
12		ontr2 5772	. . . 4 |- ( ( ( x i^i y ) e. On /\ B e. On ) -> ( ( ( x i^i y ) C_ x /\ x e. B ) -> ( x i^i y ) e. B ) )
13	7, 11, 12	syl6c 70	. . 3 |- ( B e. On -> ( ( x e. B /\ y e. B ) -> ( x i^i y ) e. B ) )
14	13	ralrimivv 2970	. 2 |- ( B e. On -> A. x e. B A. y e. B ( x i^i y ) e. B )
15		fiinbas 20756	. 2 |- ( ( B e. On /\ A. x e. B A. y e. B ( x i^i y ) e. B ) -> B e. TopBases )
16	14, 15	mpdan 702	1 |- ( B e. On -> B e. TopBases )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   i^i cin 3573   C_ wss 3574   Oncon0 5723   TopBasesctb 20749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-bases 20750

This theorem is referenced by:  onsstopbas  32428  onsuctop  32432