Theorem ntrss 7685

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 7684	. . . . . 6 |- ((J e. Top /\ (X \ T) (_ X /\ (X \ S) (_ (X \ T)) -> ((cls` J)` (X \ S)) (_ ((cls` J)` (X \ T)))
3	2	3expb 836	. . . . 5 |- ((J e. Top /\ ((X \ T) (_ X /\ (X \ S) (_ (X \ T))) -> ((cls` J)` (X \ S)) (_ ((cls` J)` (X \ T)))
4		sscon 2174	. . . . . . 7 |- (T (_ S -> (X \ S) (_ (X \ T))
5	4	adantl 390	. . . . . 6 |- ((S (_ X /\ T (_ S) -> (X \ S) (_ (X \ T))
6		difss 2170	. . . . . 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 453	. . . 4 |- ((J e. Top /\ (S (_ X /\ T (_ S)) -> ((cls` J)` (X \ S)) (_ ((cls` J)` (X \ T)))
9		sscon 2174	. . . 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 7683	. . . 4 |- ((J e. Top /\ T (_ X) -> ((int` J)` T) = (X \ ((cls` J)` (X \ T))))
12		sstr2 2074	. . . . 5 |- (T (_ S -> (S (_ X -> T (_ X))
13	12	impcom 351	. . . 4 |- ((S (_ X /\ T (_ S) -> T (_ X)
14	11, 13	sylan2 453	. . 3 |- ((J e. Top /\ (S (_ X /\ T (_ S)) -> ((int` J)` T) = (X \ ((cls` J)` (X \ T))))
15	1	ntrval2 7683	. . . 4 |- ((J e. Top /\ S (_ X) -> ((int` J)` S) = (X \ ((cls` J)` (X \ S))))
16	15	adantrr 397	. . 3 |- ((J e. Top /\ (S (_ X /\ T (_ S)) -> ((int` J)` S) = (X \ ((cls` J)` (X \ S))))
17	10, 14, 16	3sstr4d 2107	. 2 |- ((J e. Top /\ (S (_ X /\ T (_ S)) -> ((int` J)` T) (_ ((int` J)` S))
18	17	3impb 831	1 |- ((J e. Top /\ S (_ X /\ T (_ S) -> ((int` J)` T) (_ ((int` J)` S))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960   \ cdif 2047   (_ wss 2050  U.cuni 2507  ` cfv 3188  Topctop 7590  intcnt 7658  clsccl 7659

This theorem is referenced by:  ntrcls0 7704

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-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-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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