Theorem 0ntr 8978

Description: A subset with an empty interior cannot cover a whole (nonempty) topology.

Hypothesis

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

Assertion

Ref	Expression
0ntr	|- (((J e. Top /\ X =/= (/)) /\ (S C_ X /\ ((int` J)` S) = (/))) -> (X \ S) =/= (/))

Proof of Theorem 0ntr

Step	Hyp	Ref	Expression
1		fveq2 4681	. . . . . . . . . . . . 13 |- (S = X -> ((int` J)` S) = ((int` J)` X))
2		clscld.1	. . . . . . . . . . . . . 14 |- X = U.J
3	2	ntrtop 8977	. . . . . . . . . . . . 13 |- (J e. Top -> ((int` J)` X) = X)
4	1, 3	sylan9eqr 1951	. . . . . . . . . . . 12 |- ((J e. Top /\ S = X) -> ((int` J)` S) = X)
5	4	eqeq1d 1892	. . . . . . . . . . 11 |- ((J e. Top /\ S = X) -> (((int` J)` S) = (/) <-> X = (/)))
6	5	biimpd 170	. . . . . . . . . 10 |- ((J e. Top /\ S = X) -> (((int` J)` S) = (/) -> X = (/)))
7	6	ex 402	. . . . . . . . 9 |- (J e. Top -> (S = X -> (((int` J)` S) = (/) -> X = (/))))
8		eqss 2631	. . . . . . . . 9 |- (S = X <-> (S C_ X /\ X C_ S))
9	7, 8	syl5ibr 224	. . . . . . . 8 |- (J e. Top -> ((S C_ X /\ X C_ S) -> (((int` J)` S) = (/) -> X = (/))))
10	9	exp3a 405	. . . . . . 7 |- (J e. Top -> (S C_ X -> (X C_ S -> (((int` J)` S) = (/) -> X = (/)))))
11	10	com34 40	. . . . . 6 |- (J e. Top -> (S C_ X -> (((int` J)` S) = (/) -> (X C_ S -> X = (/)))))
12	11	imp32 390	. . . . 5 |- ((J e. Top /\ (S C_ X /\ ((int` J)` S) = (/))) -> (X C_ S -> X = (/)))
13		ssdif0 2934	. . . . 5 |- (X C_ S <-> (X \ S) = (/))
14	12, 13	syl5ibr 224	. . . 4 |- ((J e. Top /\ (S C_ X /\ ((int` J)` S) = (/))) -> ((X \ S) = (/) -> X = (/)))
15	14	necon3d 2041	. . 3 |- ((J e. Top /\ (S C_ X /\ ((int` J)` S) = (/))) -> (X =/= (/) -> (X \ S) =/= (/)))
16	15	imp 377	. 2 |- (((J e. Top /\ (S C_ X /\ ((int` J)` S) = (/))) /\ X =/= (/)) -> (X \ S) =/= (/))
17	16	an1rs 547	1 |- (((J e. Top /\ X =/= (/)) /\ (S C_ X /\ ((int` J)` S) = (/))) -> (X \ S) =/= (/))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017   \ cdif 2590   C_ wss 2593  (/)c0 2875  U.cuni 3177  ` cfv 3998  Topctop 8857  intcnt 8937

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-top 8861  df-ntr 8940

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