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Theorem istps3OLD 18546
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 18545 . 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 969 . . . 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 4495 . . . . . . 7  |-  ( A  e.  J  ->  ~P A  e.  _V )
7 ssexg 4457 . . . . . . 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 1007 . . . . 5  |-  ( ( J  C_  ~P A  /\  (/)  e.  J  /\  A  e.  J )  ->  J  e.  _V )
10 istopg 18527 . . . . 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 965   A.wal 1367    e. wcel 1756   A.wral 2734   _Vcvv 2991    i^i cin 3346    C_ wss 3347   (/)c0 3656   ~Pcpw 3879   <.cop 3902   U.cuni 4110   Topctop 18517   TopSpOLDctpsOLD 18519
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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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 2739  df-rex 2740  df-rab 2743  df-v 2993  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-br 4312  df-opab 4370  df-xp 4865  df-rel 4866  df-top 18522  df-topspOLD 18523
This theorem is referenced by:  istps4OLD  18547
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