Theorem istps5OLD 7610

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.

Assertion

Ref	Expression
istps5OLD	|- (<.A, J>. e. TopSp <-> ((A.x e. J x (_ A /\ (/) e. J /\ A e. J) /\ (A.x(x (_ J -> U.x e. J) /\ A.x((x (_ J /\ x =/= (/) /\ E.y e. om x ~~ y) -> |^|x e. J))))

Distinct variable groups:   x,y,A   x,J,y

Proof of Theorem istps5OLD

Step	Hyp	Ref	Expression
1		istps4OLD 7609	. 2 |- (<.A, J>. e. TopSp <-> ((J (_ P~A /\ (/) e. J /\ A e. J) /\ (A.x(x (_ J -> U.x e. J) /\ A.x((x (_ J /\ x =/= (/) /\ E.y e. om x ~~ y) -> |^|x e. J))))
2		pwssb 2760	. . . 4 |- (J (_ P~A <-> A.x e. J x (_ A)
3	2	3anbi1i 824	. . 3 |- ((J (_ P~A /\ (/) e. J /\ A e. J) <-> (A.x e. J x (_ A /\ (/) e. J /\ A e. J))
4	3	anbi1i 481	. 2 |- (((J (_ P~A /\ (/) e. J /\ A e. J) /\ (A.x(x (_ J -> U.x e. J) /\ A.x((x (_ J /\ x =/= (/) /\ E.y e. om x ~~ y) -> |^|x e. J))) <-> ((A.x e. J x (_ A /\ (/) e. J /\ A e. J) /\ (A.x(x (_ J -> U.x e. J) /\ A.x((x (_ J /\ x =/= (/) /\ E.y e. om x ~~ y) -> |^|x e. J))))
5	1, 4	bitr 173	1 |- (<.A, J>. e. TopSp <-> ((A.x e. J x (_ A /\ (/) e. J /\ A e. J) /\ (A.x(x (_ J -> U.x e. J) /\ A.x((x (_ J /\ x =/= (/) /\ E.y e. om x ~~ y) -> |^|x e. J))))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775  A.wal 954   e. wcel 958   =/= wne 1585  A.wral 1645  E.wrex 1646   (_ wss 2047  (/)c0 2280  P~cpw 2401  <.cop 2411  U.cuni 2503  |^|cint 2533   class class class wbr 2619  omcom 3131   ~~ cen 4364  TopSpctps 7589

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-reg 4593

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1o 4133  df-oadd 4135  df-er 4261  df-en 4368  df-top 7592  df-topsp 7593

Copyright terms: Public domain