Theorem ntrss 7688

Description: Subset relationship for interior.

Hypothesis

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

Assertion

Ref	Expression
ntrss	|- ((J e. Top /\ S (_ X /\ T (_ S) -> ((int` J)` T) (_ ((int` J)` S))

Proof of Theorem ntrss

Step	Hyp	Ref	Expression
1		clscld.1	. . . . . . 7 |- X = U.J
2	1	clsss 7687	. . . . . 6 |- ((J e. Top /\ (X \ T) (_ X /\ (X \ S) (_ (X \ T)) -> ((cls` J)` (X \ S)) (_ ((cls` J)` (X \ T)))
3	2	3expb 834	. . . . 5 |- ((J e. Top /\ ((X \ T) (_ X /\ (X \ S) (_ (X \ T))) -> ((cls` J)` (X \ S)) (_ ((cls` J)` (X \ T)))
4		sscon 2171	. . . . . . 7 |- (T (_ S -> (X \ S) (_ (X \ T))
5	4	adantl 388	. . . . . 6 |- ((S (_ X /\ T (_ S) -> (X \ S) (_ (X \ T))
6		difss 2167	. . . . . 6 |- (X \ T) (_ X
7	5, 6	jctil 292	. . . . 5 |- ((S (_ X /\ T (_ S) -> ((X \ T) (_ X /\ (X \ S) (_ (X \ T)))
8	3, 7	sylan2 451	. . . 4 |- ((J e. Top /\ (S (_ X /\ T (_ S)) -> ((cls` J)` (X \ S)) (_ ((cls` J)` (X \ T)))
9		sscon 2171	. . . 4 |- (((cls` J)` (X \ S)) (_ ((cls` J)` (X \ T)) -> (X \ ((cls` J)` (X \ T))) (_ (X \ ((cls` J)` (X \ S))))
10	8, 9	syl 10	. . 3 |- ((J e. Top /\ (S (_ X /\ T (_ S)) -> (X \ ((cls` J)` (X \ T))) (_ (X \ ((cls` J)` (X \ S))))
11	1	ntrval2 7686	. . . 4 |- ((J e. Top /\ T (_ X) -> ((int` J)` T) = (X \ ((cls` J)` (X \ T))))
12		sstr2 2071	. . . . 5 |- (T (_ S -> (S (_ X -> T (_ X))
13	12	impcom 351	. . . 4 |- ((S (_ X /\ T (_ S) -> T (_ X)
14	11, 13	sylan2 451	. . 3 |- ((J e. Top /\ (S (_ X /\ T (_ S)) -> ((int` J)` T) = (X \ ((cls` J)` (X \ T))))
15	1	ntrval2 7686	. . . 4 |- ((J e. Top /\ S (_ X) -> ((int` J)` S) = (X \ ((cls` J)` (X \ S))))
16	15	adantrr 395	. . 3 |- ((J e. Top /\ (S (_ X /\ T (_ S)) -> ((int` J)` S) = (X \ ((cls` J)` (X \ S))))
17	10, 14, 16	3sstr4d 2104	. 2 |- ((J e. Top /\ (S (_ X /\ T (_ S)) -> ((int` J)` T) (_ ((int` J)` S))
18	17	3impb 829	1 |- ((J e. Top /\ S (_ X /\ T (_ S) -> ((int` J)` T) (_ ((int` J)` S))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958   \ cdif 2044   (_ wss 2047  U.cuni 2503  ` cfv 3182  Topctop 7588  intcnt 7661  clsccl 7662

This theorem is referenced by:  ntrcls0 7707

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-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-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-int 2534  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-cld 7663  df-ntr 7664  df-cls 7665

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