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Theorem istps4OLD 16980
Description: A standard textbook definition of a topological space. (Contributed by NM, 19-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
istps4OLD  |-  ( <. A ,  J >.  e. 
TopSp OLD  <->  ( ( J 
C_  ~P A  /\  (/)  e.  J  /\  A  e.  J
)  /\  ( A. x ( x  C_  J  ->  U. x  e.  J
)  /\  A. x
( ( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) ) ) )
Distinct variable groups:    x, A    x, J

Proof of Theorem istps4OLD
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 istps3OLD 16979 . 2  |-  ( <. A ,  J >.  e. 
TopSp OLD  <->  ( ( 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
) ) )
2 fiint 7375 . . . 4  |-  ( A. x  e.  J  A. y  e.  J  (
x  i^i  y )  e.  J  <->  A. x ( ( x  C_  J  /\  x  =/=  (/)  /\  x  e. 
Fin )  ->  |^| x  e.  J ) )
32anbi2i 676 . . 3  |-  ( ( A. x ( x 
C_  J  ->  U. x  e.  J )  /\  A. x  e.  J  A. y  e.  J  (
x  i^i  y )  e.  J )  <->  ( A. x ( x  C_  J  ->  U. x  e.  J
)  /\  A. x
( ( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) ) )
43anbi2i 676 . 2  |-  ( ( ( 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
) )  <->  ( ( J  C_  ~P A  /\  (/) 
e.  J  /\  A  e.  J )  /\  ( A. x ( x  C_  J  ->  U. x  e.  J
)  /\  A. x
( ( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) ) ) )
51, 4bitri 241 1  |-  ( <. A ,  J >.  e. 
TopSp OLD  <->  ( ( J 
C_  ~P A  /\  (/)  e.  J  /\  A  e.  J
)  /\  ( A. x ( x  C_  J  ->  U. x  e.  J
)  /\  A. x
( ( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   A.wal 1549    e. wcel 1725    =/= wne 2598   A.wral 2697    i^i cin 3311    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   <.cop 3809   U.cuni 4007   |^|cint 4042   Fincfn 7101   TopSp OLDctpsOLD 16952
This theorem is referenced by:  istps5OLD  16981
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-fin 7105  df-top 16955  df-topspOLD 16956
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