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Theorem ontopbas 24041
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
StepHypRef Expression
1 onelon 4310 . . . . . . . 8  |-  ( ( B  e.  On  /\  x  e.  B )  ->  x  e.  On )
2 onelon 4310 . . . . . . . 8  |-  ( ( B  e.  On  /\  y  e.  B )  ->  y  e.  On )
31, 2anim12dan 813 . . . . . . 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 4316 . . . . . 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 543 . . . 4  |-  ( B  e.  On  ->  (
( x  e.  B  /\  y  e.  B
)  ->  ( (
x  i^i  y )  e.  On  /\  B  e.  On ) ) )
8 inss1 3296 . . . . . . 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 12 . . . 4  |-  ( B  e.  On  ->  (
( x  e.  B  /\  y  e.  B
)  ->  ( (
x  i^i  y )  C_  x  /\  x  e.  B ) ) )
12 ontr2 4332 . . . 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 2596 . 2  |-  ( B  e.  On  ->  A. x  e.  B  A. y  e.  B  ( x  i^i  y )  e.  B
)
15 fiinbas 16522 . 2  |-  ( ( B  e.  On  /\  A. x  e.  B  A. y  e.  B  (
x  i^i  y )  e.  B )  ->  B  e. 
TopBases )
1614, 15mpdan 652 1  |-  ( B  e.  On  ->  B  e. 
TopBases )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    e. wcel 1621   A.wral 2509    i^i cin 3077    C_ wss 3078   Oncon0 4285   TopBasesctb 16467
This theorem is referenced by:  onsstopbas  24042  onsuctop  24046
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-tr 4011  df-eprel 4198  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-bases 16470
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