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Theorem istps3OLD 19550
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. 
TopSpOLD  <->  ( ( 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 19549 . 2  |-  ( <. A ,  J >.  e. 
TopSpOLD  <->  ( ( J  e.  Top  /\  J  C_ 
~P A )  /\  ( (/)  e.  J  /\  A  e.  J )
) )
2 anass 649 . 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 450 . . 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 977 . . . 4  |-  ( ( J  C_  ~P A  /\  (/)  e.  J  /\  A  e.  J )  <->  ( J  C_  ~P A  /\  ( (/)  e.  J  /\  A  e.  J
) ) )
54anbi1i 695 . . 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 4640 . . . . . . 7  |-  ( A  e.  J  ->  ~P A  e.  _V )
7 ssexg 4602 . . . . . . 7  |-  ( ( J  C_  ~P A  /\  ~P A  e.  _V )  ->  J  e.  _V )
86, 7sylan2 474 . . . . . 6  |-  ( ( J  C_  ~P A  /\  A  e.  J
)  ->  J  e.  _V )
983adant2 1015 . . . . 5  |-  ( ( J  C_  ~P A  /\  (/)  e.  J  /\  A  e.  J )  ->  J  e.  _V )
10 istopg 19531 . . . . 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 16 . . . 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 637 . . 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 273 . 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 271 1  |-  ( <. A ,  J >.  e. 
TopSpOLD  <->  ( ( 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973   A.wal 1393    e. wcel 1819   A.wral 2807   _Vcvv 3109    i^i cin 3470    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   <.cop 4038   U.cuni 4251   Topctop 19521   TopSpOLDctpsOLD 19523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-top 19526  df-topspOLD 19527
This theorem is referenced by:  istps4OLD  19551
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