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Theorem istps2OLD 16987
Description: Express the predicate "is a topological space." (Contributed by NM, 18-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
istps2OLD  |-  ( <. A ,  J >.  e. 
TopSp OLD  <->  ( ( J  e.  Top  /\  J  C_ 
~P A )  /\  ( (/)  e.  J  /\  A  e.  J )
) )

Proof of Theorem istps2OLD
StepHypRef Expression
1 eqimss 3401 . . . . . . . 8  |-  ( U. J  =  A  ->  U. J  C_  A )
21adantl 454 . . . . . . 7  |-  ( ( J  e.  Top  /\  U. J  =  A )  ->  U. J  C_  A
)
3 0opn 16978 . . . . . . . 8  |-  ( J  e.  Top  ->  (/)  e.  J
)
43adantr 453 . . . . . . 7  |-  ( ( J  e.  Top  /\  U. J  =  A )  ->  (/)  e.  J )
5 simpr 449 . . . . . . . 8  |-  ( ( J  e.  Top  /\  U. J  =  A )  ->  U. J  =  A )
6 eqid 2437 . . . . . . . . . 10  |-  U. J  =  U. J
76topopn 16980 . . . . . . . . 9  |-  ( J  e.  Top  ->  U. J  e.  J )
87adantr 453 . . . . . . . 8  |-  ( ( J  e.  Top  /\  U. J  =  A )  ->  U. J  e.  J
)
95, 8eqeltrrd 2512 . . . . . . 7  |-  ( ( J  e.  Top  /\  U. J  =  A )  ->  A  e.  J
)
102, 4, 9jca32 523 . . . . . 6  |-  ( ( J  e.  Top  /\  U. J  =  A )  ->  ( U. J  C_  A  /\  ( (/)  e.  J  /\  A  e.  J ) ) )
1110ex 425 . . . . 5  |-  ( J  e.  Top  ->  ( U. J  =  A  ->  ( U. J  C_  A  /\  ( (/)  e.  J  /\  A  e.  J
) ) ) )
12 unissel 4045 . . . . . 6  |-  ( ( U. J  C_  A  /\  A  e.  J
)  ->  U. J  =  A )
1312adantrl 698 . . . . 5  |-  ( ( U. J  C_  A  /\  ( (/)  e.  J  /\  A  e.  J
) )  ->  U. J  =  A )
1411, 13impbid1 196 . . . 4  |-  ( J  e.  Top  ->  ( U. J  =  A  <->  ( U. J  C_  A  /\  ( (/)  e.  J  /\  A  e.  J
) ) ) )
15 eqcom 2439 . . . 4  |-  ( A  =  U. J  <->  U. J  =  A )
16 sspwuni 4177 . . . . 5  |-  ( J 
C_  ~P A  <->  U. J  C_  A )
1716anbi1i 678 . . . 4  |-  ( ( J  C_  ~P A  /\  ( (/)  e.  J  /\  A  e.  J
) )  <->  ( U. J  C_  A  /\  ( (/) 
e.  J  /\  A  e.  J ) ) )
1814, 15, 173bitr4g 281 . . 3  |-  ( J  e.  Top  ->  ( A  =  U. J  <->  ( J  C_ 
~P A  /\  ( (/) 
e.  J  /\  A  e.  J ) ) ) )
1918pm5.32i 620 . 2  |-  ( ( J  e.  Top  /\  A  =  U. J )  <-> 
( J  e.  Top  /\  ( J  C_  ~P A  /\  ( (/)  e.  J  /\  A  e.  J
) ) ) )
20 istpsOLD 16986 . 2  |-  ( <. A ,  J >.  e. 
TopSp OLD  <->  ( J  e. 
Top  /\  A  =  U. J ) )
21 anass 632 . 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 ) ) ) )
2219, 20, 213bitr4i 270 1  |-  ( <. A ,  J >.  e. 
TopSp OLD  <->  ( ( J  e.  Top  /\  J  C_ 
~P A )  /\  ( (/)  e.  J  /\  A  e.  J )
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    C_ wss 3321   (/)c0 3629   ~Pcpw 3800   <.cop 3818   U.cuni 4016   Topctop 16959   TopSp OLDctpsOLD 16961
This theorem is referenced by:  istps3OLD  16988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-xp 4885  df-rel 4886  df-top 16964  df-topspOLD 16965
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