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Theorem istps2OLD 16659
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 3230 . . . . . . . 8  |-  ( U. J  =  A  ->  U. J  C_  A )
21adantl 452 . . . . . . 7  |-  ( ( J  e.  Top  /\  U. J  =  A )  ->  U. J  C_  A
)
3 0opn 16650 . . . . . . . 8  |-  ( J  e.  Top  ->  (/)  e.  J
)
43adantr 451 . . . . . . 7  |-  ( ( J  e.  Top  /\  U. J  =  A )  ->  (/)  e.  J )
5 simpr 447 . . . . . . . 8  |-  ( ( J  e.  Top  /\  U. J  =  A )  ->  U. J  =  A )
6 eqid 2283 . . . . . . . . . 10  |-  U. J  =  U. J
76topopn 16652 . . . . . . . . 9  |-  ( J  e.  Top  ->  U. J  e.  J )
87adantr 451 . . . . . . . 8  |-  ( ( J  e.  Top  /\  U. J  =  A )  ->  U. J  e.  J
)
95, 8eqeltrrd 2358 . . . . . . 7  |-  ( ( J  e.  Top  /\  U. J  =  A )  ->  A  e.  J
)
102, 4, 9jca32 521 . . . . . 6  |-  ( ( J  e.  Top  /\  U. J  =  A )  ->  ( U. J  C_  A  /\  ( (/)  e.  J  /\  A  e.  J ) ) )
1110ex 423 . . . . 5  |-  ( J  e.  Top  ->  ( U. J  =  A  ->  ( U. J  C_  A  /\  ( (/)  e.  J  /\  A  e.  J
) ) ) )
12 unissel 3856 . . . . . 6  |-  ( ( U. J  C_  A  /\  A  e.  J
)  ->  U. J  =  A )
1312adantrl 696 . . . . 5  |-  ( ( U. J  C_  A  /\  ( (/)  e.  J  /\  A  e.  J
) )  ->  U. J  =  A )
1411, 13impbid1 194 . . . 4  |-  ( J  e.  Top  ->  ( U. J  =  A  <->  ( U. J  C_  A  /\  ( (/)  e.  J  /\  A  e.  J
) ) ) )
15 eqcom 2285 . . . 4  |-  ( A  =  U. J  <->  U. J  =  A )
16 sspwuni 3987 . . . . 5  |-  ( J 
C_  ~P A  <->  U. J  C_  A )
1716anbi1i 676 . . . 4  |-  ( ( J  C_  ~P A  /\  ( (/)  e.  J  /\  A  e.  J
) )  <->  ( U. J  C_  A  /\  ( (/) 
e.  J  /\  A  e.  J ) ) )
1814, 15, 173bitr4g 279 . . 3  |-  ( J  e.  Top  ->  ( A  =  U. J  <->  ( J  C_ 
~P A  /\  ( (/) 
e.  J  /\  A  e.  J ) ) ) )
1918pm5.32i 618 . 2  |-  ( ( J  e.  Top  /\  A  =  U. J )  <-> 
( J  e.  Top  /\  ( J  C_  ~P A  /\  ( (/)  e.  J  /\  A  e.  J
) ) ) )
20 istpsOLD 16658 . 2  |-  ( <. A ,  J >.  e. 
TopSp OLD  <->  ( J  e. 
Top  /\  A  =  U. J ) )
21 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 ) ) ) )
2219, 20, 213bitr4i 268 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 176    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   <.cop 3643   U.cuni 3827   Topctop 16631   TopSp OLDctpsOLD 16633
This theorem is referenced by:  istps3OLD  16660
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-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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