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Theorem istps2OLD 16715
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 3264 . . . . . . . 8  |-  ( U. J  =  A  ->  U. J  C_  A )
21adantl 452 . . . . . . 7  |-  ( ( J  e.  Top  /\  U. J  =  A )  ->  U. J  C_  A
)
3 0opn 16706 . . . . . . . 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 2316 . . . . . . . . . 10  |-  U. J  =  U. J
76topopn 16708 . . . . . . . . 9  |-  ( J  e.  Top  ->  U. J  e.  J )
87adantr 451 . . . . . . . 8  |-  ( ( J  e.  Top  /\  U. J  =  A )  ->  U. J  e.  J
)
95, 8eqeltrrd 2391 . . . . . . 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 3893 . . . . . 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 2318 . . . 4  |-  ( A  =  U. J  <->  U. J  =  A )
16 sspwuni 4024 . . . . 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 16714 . 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 1633    e. wcel 1701    C_ wss 3186   (/)c0 3489   ~Pcpw 3659   <.cop 3677   U.cuni 3864   Topctop 16687   TopSp OLDctpsOLD 16689
This theorem is referenced by:  istps3OLD  16716
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-xp 4732  df-rel 4733  df-top 16692  df-topspOLD 16693
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