Theorem clsndisj 7703

Description: Any open set containing a point that belongs to the closure of a subset intersects the subset. One direction of Theorem 6.5(a) of [Munkres] p. 95.

Hypothesis

Ref	Expression
clscld.1	|- X = U.J

Assertion

Ref	Expression
clsndisj	|- (((J e. Top /\ S (_ X /\ P e. ((cls` J)` S)) /\ (U e. J /\ P e. U)) -> (U i^i S) =/= (/))

Proof of Theorem clsndisj

Step	Hyp	Ref	Expression
1		eleq2 1538	. . . . 5 |- (x = U -> (P e. x <-> P e. U))
2		ineq1 2213	. . . . . 6 |- (x = U -> (x i^i S) = (U i^i S))
3	2	neeq1d 1597	. . . . 5 |- (x = U -> ((x i^i S) =/= (/) <-> (U i^i S) =/= (/)))
4	1, 3	imbi12d 628	. . . 4 |- (x = U -> ((P e. x -> (x i^i S) =/= (/)) <-> (P e. U -> (U i^i S) =/= (/))))
5	4	rcla4cv 1877	. . 3 |- (A.x e. J (P e. x -> (x i^i S) =/= (/)) -> (U e. J -> (P e. U -> (U i^i S) =/= (/))))
6	5	imp32 363	. 2 |- ((A.x e. J (P e. x -> (x i^i S) =/= (/)) /\ (U e. J /\ P e. U)) -> (U i^i S) =/= (/))
7		clscld.1	. . . . 5 |- X = U.J
8	7	elcls 7701	. . . 4 |- ((J e. Top /\ S (_ X /\ P e. X) -> (P e. ((cls` J)` S) <-> A.x e. J (P e. x -> (x i^i S) =/= (/))))
9	8	biimpa 418	. . 3 |- (((J e. Top /\ S (_ X /\ P e. X) /\ P e. ((cls` J)` S)) -> A.x e. J (P e. x -> (x i^i S) =/= (/)))
10		3simp1 790	. . . 4 |- ((J e. Top /\ S (_ X /\ P e. ((cls` J)` S)) -> J e. Top)
11		3simp2 791	. . . 4 |- ((J e. Top /\ S (_ X /\ P e. ((cls` J)` S)) -> S (_ X)
12	7	clsss3 7688	. . . . . 6 |- ((J e. Top /\ S (_ X) -> ((cls` J)` S) (_ X)
13	12	sseld 2070	. . . . 5 |- ((J e. Top /\ S (_ X) -> (P e. ((cls` J)` S) -> P e. X))
14	13	3impia 832	. . . 4 |- ((J e. Top /\ S (_ X /\ P e. ((cls` J)` S)) -> P e. X)
15	10, 11, 14	3jca 821	. . 3 |- ((J e. Top /\ S (_ X /\ P e. ((cls` J)` S)) -> (J e. Top /\ S (_ X /\ P e. X))
16		3simp3 792	. . 3 |- ((J e. Top /\ S (_ X /\ P e. ((cls` J)` S)) -> P e. ((cls` J)` S))
17	9, 15, 16	sylanc 473	. 2 |- ((J e. Top /\ S (_ X /\ P e. ((cls` J)` S)) -> A.x e. J (P e. x -> (x i^i S) =/= (/)))
18	6, 17	sylan 450	1 |- (((J e. Top /\ S (_ X /\ P e. ((cls` J)` S)) /\ (U e. J /\ P e. U)) -> (U i^i S) =/= (/))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960   =/= wne 1588  A.wral 1648   i^i cin 2049   (_ wss 2050  (/)c0 2283  U.cuni 2507  ` cfv 3188  Topctop 7590  clsccl 7659

This theorem is referenced by:  neindisj 7728

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-iin 2573  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-top 7594  df-cld 7660  df-ntr 7661  df-cls 7662

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