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Theorem istps5OLD 18529
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. 
TopSpOLD  <->  ( ( 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 18528 . 2  |-  ( <. A ,  J >.  e. 
TopSpOLD  <->  ( ( 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 4257 . . . 4  |-  ( J 
C_  ~P A  <->  A. x  e.  J  x  C_  A
)
323anbi1i 1178 . . 3  |-  ( ( J  C_  ~P A  /\  (/)  e.  J  /\  A  e.  J )  <->  ( A. x  e.  J  x  C_  A  /\  (/)  e.  J  /\  A  e.  J
) )
43anbi1i 695 . 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 249 1  |-  ( <. A ,  J >.  e. 
TopSpOLD  <->  ( ( 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965   A.wal 1367    e. wcel 1756    =/= wne 2606   A.wral 2715    C_ wss 3328   (/)c0 3637   ~Pcpw 3860   <.cop 3883   U.cuni 4091   |^|cint 4128   Fincfn 7310   TopSpOLDctpsOLD 18500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-en 7311  df-fin 7314  df-top 18503  df-topspOLD 18504
This theorem is referenced by: (None)
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