Users' Mathboxes Mathbox for Chen-Pang He < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ontopbas Structured version   Unicode version

Theorem ontopbas 26209
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.
StepHypRef Expression
1 onelon 4635 . . . . . . . 8  |-  ( ( B  e.  On  /\  x  e.  B )  ->  x  e.  On )
2 onelon 4635 . . . . . . . 8  |-  ( ( B  e.  On  /\  y  e.  B )  ->  y  e.  On )
31, 2anim12dan 812 . . . . . . 7  |-  ( ( B  e.  On  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x  e.  On  /\  y  e.  On )
)
43ex 425 . . . . . 6  |-  ( B  e.  On  ->  (
( x  e.  B  /\  y  e.  B
)  ->  ( x  e.  On  /\  y  e.  On ) ) )
5 onin 4641 . . . . . 6  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  i^i  y
)  e.  On )
64, 5syl6 32 . . . . 5  |-  ( B  e.  On  ->  (
( x  e.  B  /\  y  e.  B
)  ->  ( x  i^i  y )  e.  On ) )
76anc2ri 543 . . . 4  |-  ( B  e.  On  ->  (
( x  e.  B  /\  y  e.  B
)  ->  ( (
x  i^i  y )  e.  On  /\  B  e.  On ) ) )
8 inss1 3546 . . . . . . 7  |-  ( x  i^i  y )  C_  x
98jctl 527 . . . . . 6  |-  ( x  e.  B  ->  (
( x  i^i  y
)  C_  x  /\  x  e.  B )
)
109adantr 453 . . . . 5  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( ( x  i^i  y )  C_  x  /\  x  e.  B
) )
1110a1i 11 . . . 4  |-  ( B  e.  On  ->  (
( x  e.  B  /\  y  e.  B
)  ->  ( (
x  i^i  y )  C_  x  /\  x  e.  B ) ) )
12 ontr2 4657 . . . 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
) )
137, 11, 12syl6c 63 . . 3  |-  ( B  e.  On  ->  (
( x  e.  B  /\  y  e.  B
)  ->  ( x  i^i  y )  e.  B
) )
1413ralrimivv 2803 . 2  |-  ( B  e.  On  ->  A. x  e.  B  A. y  e.  B  ( x  i^i  y )  e.  B
)
15 fiinbas 17048 . 2  |-  ( ( B  e.  On  /\  A. x  e.  B  A. y  e.  B  (
x  i^i  y )  e.  B )  ->  B  e. 
TopBases )
1614, 15mpdan 651 1  |-  ( B  e.  On  ->  B  e. 
TopBases )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1727   A.wral 2711    i^i cin 3305    C_ wss 3306   Oncon0 4610   TopBasesctb 16993
This theorem is referenced by:  onsstopbas  26210  onsuctop  26214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-tr 4328  df-eprel 4523  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-bases 16996
  Copyright terms: Public domain W3C validator