Theorem 0ntr 7702

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 (_ X /\ ((int` J)` S) = (/))) -> (X \ S) =/= (/))

Proof of Theorem 0ntr

Step	Hyp	Ref	Expression
1		fveq2 3724	. . . . . . . . . . . . 13 |- (S = X -> ((int` J)` S) = ((int` J)` X))
2		clscld.1	. . . . . . . . . . . . . 14 |- X = U.J
3	2	ntrtop 7701	. . . . . . . . . . . . 13 |- (J e. Top -> ((int` J)` X) = X)
4	1, 3	sylan9eqr 1529	. . . . . . . . . . . 12 |- ((J e. Top /\ S = X) -> ((int` J)` S) = X)
5	4	eqeq1d 1483	. . . . . . . . . . 11 |- ((J e. Top /\ S = X) -> (((int` J)` S) = (/) <-> X = (/)))
6	5	biimpd 153	. . . . . . . . . 10 |- ((J e. Top /\ S = X) -> (((int` J)` S) = (/) -> X = (/)))
7	6	ex 373	. . . . . . . . 9 |- (J e. Top -> (S = X -> (((int` J)` S) = (/) -> X = (/))))
8		eqss 2077	. . . . . . . . 9 |- (S = X <-> (S (_ X /\ X (_ S))
9	7, 8	syl5ibr 207	. . . . . . . 8 |- (J e. Top -> ((S (_ X /\ X (_ S) -> (((int` J)` S) = (/) -> X = (/))))
10	9	exp3a 375	. . . . . . 7 |- (J e. Top -> (S (_ X -> (X (_ S -> (((int` J)` S) = (/) -> X = (/)))))
11	10	com34 36	. . . . . 6 |- (J e. Top -> (S (_ X -> (((int` J)` S) = (/) -> (X (_ S -> X = (/)))))
12	11	imp32 363	. . . . 5 |- ((J e. Top /\ (S (_ X /\ ((int` J)` S) = (/))) -> (X (_ S -> X = (/)))
13		ssdif0 2327	. . . . 5 |- (X (_ S <-> (X \ S) = (/))
14	12, 13	syl5ibr 207	. . . 4 |- ((J e. Top /\ (S (_ X /\ ((int` J)` S) = (/))) -> ((X \ S) = (/) -> X = (/)))
15	14	necon3d 1604	. . 3 |- ((J e. Top /\ (S (_ X /\ ((int` J)` S) = (/))) -> (X =/= (/) -> (X \ S) =/= (/)))
16	15	imp 350	. 2 |- (((J e. Top /\ (S (_ X /\ ((int` J)` S) = (/))) /\ X =/= (/)) -> (X \ S) =/= (/))
17	16	an1rs 489	1 |- (((J e. Top /\ X =/= (/)) /\ (S (_ X /\ ((int` J)` S) = (/))) -> (X \ S) =/= (/))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958   =/= wne 1585   \ cdif 2044   (_ wss 2047  (/)c0 2280  U.cuni 2503  ` cfv 3182  Topctop 7588  intcnt 7661

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-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  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-fv 3198  df-top 7592  df-ntr 7664

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