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Theorem istps5OLD 16989
Description: A standard textbook definition of a topological space 
<. A ,  J >.: a topology on  A is a collection  J of subsets of  A such that  (/) and  A are in  J and that is closed under union and finite intersection. Definition of topological space in [Munkres] p. 76. (Contributed by NM, 19-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
istps5OLD  |-  ( <. A ,  J >.  e. 
TopSp OLD  <->  ( ( A. x  e.  J  x  C_  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 istps5OLD
StepHypRef Expression
1 istps4OLD 16988 . 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
( ( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) ) ) )
2 pwssb 4177 . . . 4  |-  ( J 
C_  ~P A  <->  A. x  e.  J  x  C_  A
)
323anbi1i 1144 . . 3  |-  ( ( J  C_  ~P A  /\  (/)  e.  J  /\  A  e.  J )  <->  ( A. x  e.  J  x  C_  A  /\  (/)  e.  J  /\  A  e.  J
) )
43anbi1i 677 . 2  |-  ( ( ( 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 ) ) )  <-> 
( ( A. x  e.  J  x  C_  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  <->  ( ( A. x  e.  J  x  C_  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 2599   A.wral 2705    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   <.cop 3817   U.cuni 4015   |^|cint 4050   Fincfn 7109   TopSp OLDctpsOLD 16960
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-fin 7113  df-top 16963  df-topspOLD 16964
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