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

Theorem ontopbas 26126
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 4598 . . . . . . . 8  |-  ( ( B  e.  On  /\  x  e.  B )  ->  x  e.  On )
2 onelon 4598 . . . . . . . 8  |-  ( ( B  e.  On  /\  y  e.  B )  ->  y  e.  On )
31, 2anim12dan 811 . . . . . . 7  |-  ( ( B  e.  On  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x  e.  On  /\  y  e.  On )
)
43ex 424 . . . . . 6  |-  ( B  e.  On  ->  (
( x  e.  B  /\  y  e.  B
)  ->  ( x  e.  On  /\  y  e.  On ) ) )
5 onin 4604 . . . . . 6  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  i^i  y
)  e.  On )
64, 5syl6 31 . . . . 5  |-  ( B  e.  On  ->  (
( x  e.  B  /\  y  e.  B
)  ->  ( x  i^i  y )  e.  On ) )
76anc2ri 542 . . . 4  |-  ( B  e.  On  ->  (
( x  e.  B  /\  y  e.  B
)  ->  ( (
x  i^i  y )  e.  On  /\  B  e.  On ) ) )
8 inss1 3553 . . . . . . 7  |-  ( x  i^i  y )  C_  x
98jctl 526 . . . . . 6  |-  ( x  e.  B  ->  (
( x  i^i  y
)  C_  x  /\  x  e.  B )
)
109adantr 452 . . . . 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 4620 . . . 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 62 . . 3  |-  ( B  e.  On  ->  (
( x  e.  B  /\  y  e.  B
)  ->  ( x  i^i  y )  e.  B
) )
1413ralrimivv 2789 . 2  |-  ( B  e.  On  ->  A. x  e.  B  A. y  e.  B  ( x  i^i  y )  e.  B
)
15 fiinbas 17005 . 2  |-  ( ( B  e.  On  /\  A. x  e.  B  A. y  e.  B  (
x  i^i  y )  e.  B )  ->  B  e. 
TopBases )
1614, 15mpdan 650 1  |-  ( B  e.  On  ->  B  e. 
TopBases )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   A.wral 2697    i^i cin 3311    C_ wss 3312   Oncon0 4573   TopBasesctb 16950
This theorem is referenced by:  onsstopbas  26127  onsuctop  26131
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-bases 16953
  Copyright terms: Public domain W3C validator