Theorem bcthlem10 8008

Description: Lemma for bcth 8032. If M is rare in X, the complement of its closure is not empty and is open.

Hypotheses

Ref	Expression
bcthlem6.1	|- D e. CMet
bcthlem6.3	|- X = dom dom D
bcthlem6.4	|- J = (Open` D)
bcthlem11.2	|- X =/= (/)

Assertion

Ref	Expression
bcthlem10	|- ((M (_ X /\ ((int` J)` ((cls` J)` M)) = (/)) -> ((X \ ((cls` J)` M)) =/= (/) /\ (X \ ((cls` J)` M)) e. J))

Proof of Theorem bcthlem10

Step	Hyp	Ref	Expression
1		bcthlem6.1	. . . . . . . . 9 |- D e. CMet
2		bcthlem6.3	. . . . . . . . 9 |- X = dom dom D
3		bcthlem6.4	. . . . . . . . 9 |- J = (Open` D)
4	1, 2, 3	bcthlem6 8004	. . . . . . . 8 |- J e. Top
5	1, 2, 3	bcthlem5 8003	. . . . . . . . 9 |- X = U.J
6	5	clsss3 7691	. . . . . . . 8 |- ((J e. Top /\ M (_ X) -> ((cls` J)` M) (_ X)
7	4, 6	mpan 695	. . . . . . 7 |- (M (_ X -> ((cls` J)` M) (_ X)
8		pm3.26 319	. . . . . . . . 9 |- ((((cls` J)` M) (_ X /\ X (_ ((cls` J)` M)) -> ((cls` J)` M) (_ X)
9		pm3.27 323	. . . . . . . . 9 |- ((((cls` J)` M) (_ X /\ X (_ ((cls` J)` M)) -> X (_ ((cls` J)` M))
10	8, 9	eqssd 2079	. . . . . . . 8 |- ((((cls` J)` M) (_ X /\ X (_ ((cls` J)` M)) -> ((cls` J)` M) = X)
11	10	ex 373	. . . . . . 7 |- (((cls` J)` M) (_ X -> (X (_ ((cls` J)` M) -> ((cls` J)` M) = X))
12	7, 11	syl 10	. . . . . 6 |- (M (_ X -> (X (_ ((cls` J)` M) -> ((cls` J)` M) = X))
13		ssdif0 2327	. . . . . 6 |- (X (_ ((cls` J)` M) <-> (X \ ((cls` J)` M)) = (/))
14	12, 13	syl5ibr 207	. . . . 5 |- (M (_ X -> ((X \ ((cls` J)` M)) = (/) -> ((cls` J)` M) = X))
15	14	necon3d 1604	. . . 4 |- (M (_ X -> (((cls` J)` M) =/= X -> (X \ ((cls` J)` M)) =/= (/)))
16	15	imp 350	. . 3 |- ((M (_ X /\ ((cls` J)` M) =/= X) -> (X \ ((cls` J)` M)) =/= (/))
17		bcthlem11.2	. . . . 5 |- X =/= (/)
18		fveq2 3724	. . . . . . 7 |- (((cls` J)` M) = X -> ((int` J)` ((cls` J)` M)) = ((int` J)` X))
19	5	ntrtop 7701	. . . . . . . 8 |- (J e. Top -> ((int` J)` X) = X)
20	4, 19	ax-mp 7	. . . . . . 7 |- ((int` J)` X) = X
21	18, 20	syl6req 1524	. . . . . 6 |- (((cls` J)` M) = X -> X = ((int` J)` ((cls` J)` M)))
22	21	neeq1d 1594	. . . . 5 |- (((cls` J)` M) = X -> (X =/= (/) <-> ((int` J)` ((cls` J)` M)) =/= (/)))
23	17, 22	mpbii 193	. . . 4 |- (((cls` J)` M) = X -> ((int` J)` ((cls` J)` M)) =/= (/))
24	23	necon2i 1613	. . 3 |- (((int` J)` ((cls` J)` M)) = (/) -> ((cls` J)` M) =/= X)
25	16, 24	sylan2 451	. 2 |- ((M (_ X /\ ((int` J)` ((cls` J)` M)) = (/)) -> (X \ ((cls` J)` M)) =/= (/))
26	5	cmclsopn 7693	. . . 4 |- ((J e. Top /\ M (_ X) -> (X \ ((cls` J)` M)) e. J)
27	4, 26	mpan 695	. . 3 |- (M (_ X -> (X \ ((cls` J)` M)) e. J)
28	27	adantr 389	. 2 |- ((M (_ X /\ ((int` J)` ((cls` J)` M)) = (/)) -> (X \ ((cls` J)` M)) e. J)
29	25, 28	jca 288	1 |- ((M (_ X /\ ((int` J)` ((cls` J)` M)) = (/)) -> ((X \ ((cls` J)` M)) =/= (/) /\ (X \ ((cls` J)` M)) e. J))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958   =/= wne 1585   \ cdif 2044   (_ wss 2047  (/)c0 2280  dom cdm 3170  ` cfv 3182  Topctop 7588  intcnt 7661  clsccl 7662  Opencopn 7792  CMetcms 7921

This theorem is referenced by:  bcthlem33 8031

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-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  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-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-iin 2569  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  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-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom 4370  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-lt 5247  df-sub 5356  df-neg 5358  df-pnf 5487  df-mnf 5488  df-xr 5489  df-ltxr 5490  df-le 5491  df-top 7592  df-cld 7663  df-ntr 7664  df-cls 7665  df-met 7793  df-bl 7795  df-opn 7796  df-cmet 7924

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