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Theorem istps3OLD 19292
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 19291 . 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 4637 . . . . . . 7  |-  ( A  e.  J  ->  ~P A  e.  _V )
7 ssexg 4599 . . . . . . 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 19273 . . . . 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 1377    e. wcel 1767   A.wral 2817   _Vcvv 3118    i^i cin 3480    C_ wss 3481   (/)c0 3790   ~Pcpw 4016   <.cop 4039   U.cuni 4251   Topctop 19263   TopSpOLDctpsOLD 19265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-xp 5011  df-rel 5012  df-top 19268  df-topspOLD 19269
This theorem is referenced by:  istps4OLD  19293
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