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Theorem istps3OLD 16660
Description: A standard textbook definition of a topological space. (Contributed by NM, 18-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
istps3OLD  |-  ( <. A ,  J >.  e. 
TopSp OLD  <->  ( ( J 
C_  ~P A  /\  (/)  e.  J  /\  A  e.  J
)  /\  ( A. x ( x  C_  J  ->  U. x  e.  J
)  /\  A. x  e.  J  A. y  e.  J  ( x  i^i  y )  e.  J
) ) )
Distinct variable groups:    x, y, A    x, J, y

Proof of Theorem istps3OLD
StepHypRef Expression
1 istps2OLD 16659 . 2  |-  ( <. A ,  J >.  e. 
TopSp OLD  <->  ( ( J  e.  Top  /\  J  C_ 
~P A )  /\  ( (/)  e.  J  /\  A  e.  J )
) )
2 anass 630 . 2  |-  ( ( ( J  e.  Top  /\  J  C_  ~P A
)  /\  ( (/)  e.  J  /\  A  e.  J
) )  <->  ( J  e.  Top  /\  ( J 
C_  ~P A  /\  ( (/) 
e.  J  /\  A  e.  J ) ) ) )
3 ancom 437 . . 3  |-  ( ( J  e.  Top  /\  ( J  C_  ~P A  /\  ( (/)  e.  J  /\  A  e.  J
) ) )  <->  ( ( J  C_  ~P A  /\  ( (/)  e.  J  /\  A  e.  J )
)  /\  J  e.  Top ) )
4 3anass 938 . . . 4  |-  ( ( J  C_  ~P A  /\  (/)  e.  J  /\  A  e.  J )  <->  ( J  C_  ~P A  /\  ( (/)  e.  J  /\  A  e.  J
) ) )
54anbi1i 676 . . 3  |-  ( ( ( J  C_  ~P A  /\  (/)  e.  J  /\  A  e.  J )  /\  J  e.  Top ) 
<->  ( ( J  C_  ~P A  /\  ( (/) 
e.  J  /\  A  e.  J ) )  /\  J  e.  Top )
)
6 pwexg 4194 . . . . . . 7  |-  ( A  e.  J  ->  ~P A  e.  _V )
7 ssexg 4160 . . . . . . 7  |-  ( ( J  C_  ~P A  /\  ~P A  e.  _V )  ->  J  e.  _V )
86, 7sylan2 460 . . . . . 6  |-  ( ( J  C_  ~P A  /\  A  e.  J
)  ->  J  e.  _V )
983adant2 974 . . . . 5  |-  ( ( J  C_  ~P A  /\  (/)  e.  J  /\  A  e.  J )  ->  J  e.  _V )
10 istopg 16641 . . . . 5  |-  ( J  e.  _V  ->  ( J  e.  Top  <->  ( A. x ( x  C_  J  ->  U. x  e.  J
)  /\  A. x  e.  J  A. y  e.  J  ( x  i^i  y )  e.  J
) ) )
119, 10syl 15 . . . 4  |-  ( ( J  C_  ~P A  /\  (/)  e.  J  /\  A  e.  J )  ->  ( J  e.  Top  <->  ( A. x ( x  C_  J  ->  U. x  e.  J
)  /\  A. x  e.  J  A. y  e.  J  ( x  i^i  y )  e.  J
) ) )
1211pm5.32i 618 . . 3  |-  ( ( ( J  C_  ~P A  /\  (/)  e.  J  /\  A  e.  J )  /\  J  e.  Top ) 
<->  ( ( J  C_  ~P A  /\  (/)  e.  J  /\  A  e.  J
)  /\  ( A. x ( x  C_  J  ->  U. x  e.  J
)  /\  A. x  e.  J  A. y  e.  J  ( x  i^i  y )  e.  J
) ) )
133, 5, 123bitr2i 264 . 2  |-  ( ( J  e.  Top  /\  ( J  C_  ~P A  /\  ( (/)  e.  J  /\  A  e.  J
) ) )  <->  ( ( J  C_  ~P A  /\  (/) 
e.  J  /\  A  e.  J )  /\  ( A. x ( x  C_  J  ->  U. x  e.  J
)  /\  A. x  e.  J  A. y  e.  J  ( x  i^i  y )  e.  J
) ) )
141, 2, 133bitri 262 1  |-  ( <. A ,  J >.  e. 
TopSp OLD  <->  ( ( J 
C_  ~P A  /\  (/)  e.  J  /\  A  e.  J
)  /\  ( A. x ( x  C_  J  ->  U. x  e.  J
)  /\  A. x  e.  J  A. y  e.  J  ( x  i^i  y )  e.  J
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1527    e. wcel 1684   A.wral 2543   _Vcvv 2788    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   <.cop 3643   U.cuni 3827   Topctop 16631   TopSp OLDctpsOLD 16633
This theorem is referenced by:  istps4OLD  16661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-top 16636  df-topspOLD 16637
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