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Theorem ontopbas 28296
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 4765 . . . . . . . 8  |-  ( ( B  e.  On  /\  x  e.  B )  ->  x  e.  On )
2 onelon 4765 . . . . . . . 8  |-  ( ( B  e.  On  /\  y  e.  B )  ->  y  e.  On )
31, 2anim12dan 833 . . . . . . 7  |-  ( ( B  e.  On  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x  e.  On  /\  y  e.  On )
)
43ex 434 . . . . . 6  |-  ( B  e.  On  ->  (
( x  e.  B  /\  y  e.  B
)  ->  ( x  e.  On  /\  y  e.  On ) ) )
5 onin 4771 . . . . . 6  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  i^i  y
)  e.  On )
64, 5syl6 33 . . . . 5  |-  ( B  e.  On  ->  (
( x  e.  B  /\  y  e.  B
)  ->  ( x  i^i  y )  e.  On ) )
76anc2ri 558 . . . 4  |-  ( B  e.  On  ->  (
( x  e.  B  /\  y  e.  B
)  ->  ( (
x  i^i  y )  e.  On  /\  B  e.  On ) ) )
8 inss1 3591 . . . . . . 7  |-  ( x  i^i  y )  C_  x
98jctl 541 . . . . . 6  |-  ( x  e.  B  ->  (
( x  i^i  y
)  C_  x  /\  x  e.  B )
)
109adantr 465 . . . . 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 4787 . . . 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 64 . . 3  |-  ( B  e.  On  ->  (
( x  e.  B  /\  y  e.  B
)  ->  ( x  i^i  y )  e.  B
) )
1413ralrimivv 2828 . 2  |-  ( B  e.  On  ->  A. x  e.  B  A. y  e.  B  ( x  i^i  y )  e.  B
)
15 fiinbas 18579 . 2  |-  ( ( B  e.  On  /\  A. x  e.  B  A. y  e.  B  (
x  i^i  y )  e.  B )  ->  B  e. 
TopBases )
1614, 15mpdan 668 1  |-  ( B  e.  On  ->  B  e. 
TopBases )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756   A.wral 2736    i^i cin 3348    C_ wss 3349   Oncon0 4740   TopBasesctb 18524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-tr 4407  df-eprel 4653  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-bases 18527
This theorem is referenced by:  onsstopbas  28297  onsuctop  28301
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