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Theorem ontopbas 24939
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 4433 . . . . . . . 8  |-  ( ( B  e.  On  /\  x  e.  B )  ->  x  e.  On )
2 onelon 4433 . . . . . . . 8  |-  ( ( B  e.  On  /\  y  e.  B )  ->  y  e.  On )
31, 2anim12dan 810 . . . . . . 7  |-  ( ( B  e.  On  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x  e.  On  /\  y  e.  On )
)
43ex 423 . . . . . 6  |-  ( B  e.  On  ->  (
( x  e.  B  /\  y  e.  B
)  ->  ( x  e.  On  /\  y  e.  On ) ) )
5 onin 4439 . . . . . 6  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  i^i  y
)  e.  On )
64, 5syl6 29 . . . . 5  |-  ( B  e.  On  ->  (
( x  e.  B  /\  y  e.  B
)  ->  ( x  i^i  y )  e.  On ) )
76anc2ri 541 . . . 4  |-  ( B  e.  On  ->  (
( x  e.  B  /\  y  e.  B
)  ->  ( (
x  i^i  y )  e.  On  /\  B  e.  On ) ) )
8 inss1 3402 . . . . . . 7  |-  ( x  i^i  y )  C_  x
98jctl 525 . . . . . 6  |-  ( x  e.  B  ->  (
( x  i^i  y
)  C_  x  /\  x  e.  B )
)
109adantr 451 . . . . 5  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( ( x  i^i  y )  C_  x  /\  x  e.  B
) )
1110a1i 10 . . . 4  |-  ( B  e.  On  ->  (
( x  e.  B  /\  y  e.  B
)  ->  ( (
x  i^i  y )  C_  x  /\  x  e.  B ) ) )
12 ontr2 4455 . . . 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 60 . . 3  |-  ( B  e.  On  ->  (
( x  e.  B  /\  y  e.  B
)  ->  ( x  i^i  y )  e.  B
) )
1413ralrimivv 2647 . 2  |-  ( B  e.  On  ->  A. x  e.  B  A. y  e.  B  ( x  i^i  y )  e.  B
)
15 fiinbas 16706 . 2  |-  ( ( B  e.  On  /\  A. x  e.  B  A. y  e.  B  (
x  i^i  y )  e.  B )  ->  B  e. 
TopBases )
1614, 15mpdan 649 1  |-  ( B  e.  On  ->  B  e. 
TopBases )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   A.wral 2556    i^i cin 3164    C_ wss 3165   Oncon0 4408   TopBasesctb 16651
This theorem is referenced by:  onsstopbas  24940  onsuctop  24944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-bases 16654
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