Theorem ntrcls0 7704

Description: A subset whose closure has an empty interior also has an empty interior.

Hypothesis

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

Assertion

Ref	Expression
ntrcls0	|- ((J e. Top /\ S (_ X /\ ((int` J)` ((cls` J)` S)) = (/)) -> ((int` J)` S) = (/))

Proof of Theorem ntrcls0

Step	Hyp	Ref	Expression
1		clscld.1	. . . . . 6 |- X = U.J
2	1	ntrss 7685	. . . . 5 |- ((J e. Top /\ ((cls` J)` S) (_ X /\ S (_ ((cls` J)` S)) -> ((int` J)` S) (_ ((int` J)` ((cls` J)` S)))
3		pm3.26 319	. . . . 5 |- ((J e. Top /\ S (_ X) -> J e. Top)
4	1	clsss3 7688	. . . . 5 |- ((J e. Top /\ S (_ X) -> ((cls` J)` S) (_ X)
5	1	sscls 7686	. . . . 5 |- ((J e. Top /\ S (_ X) -> S (_ ((cls` J)` S))
6	2, 3, 4, 5	syl3anc 860	. . . 4 |- ((J e. Top /\ S (_ X) -> ((int` J)` S) (_ ((int` J)` ((cls` J)` S)))
7	6	3adant3 801	. . 3 |- ((J e. Top /\ S (_ X /\ ((int` J)` ((cls` J)` S)) = (/)) -> ((int` J)` S) (_ ((int` J)` ((cls` J)` S)))
8		sseq2 2086	. . . 4 |- (((int` J)` ((cls` J)` S)) = (/) -> (((int` J)` S) (_ ((int` J)` ((cls` J)` S)) <-> ((int` J)` S) (_ (/)))
9	8	3ad2ant3 804	. . 3 |- ((J e. Top /\ S (_ X /\ ((int` J)` ((cls` J)` S)) = (/)) -> (((int` J)` S) (_ ((int` J)` ((cls` J)` S)) <-> ((int` J)` S) (_ (/)))
10	7, 9	mpbid 195	. 2 |- ((J e. Top /\ S (_ X /\ ((int` J)` ((cls` J)` S)) = (/)) -> ((int` J)` S) (_ (/))
11		ss0 2307	. 2 |- (((int` J)` S) (_ (/) -> ((int` J)` S) = (/))
12	10, 11	syl 10	1 |- ((J e. Top /\ S (_ X /\ ((int` J)` ((cls` J)` S)) = (/)) -> ((int` J)` S) = (/))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960   (_ wss 2050  (/)c0 2283  U.cuni 2507  ` cfv 3188  Topctop 7590  intcnt 7658  clsccl 7659

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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