Theorem ntrcls0 21791

Description: A subset whose closure has an empty interior also has an empty interior. (Contributed by NM, 4-Oct-2007.)

Hypothesis

Ref	Expression
clscld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
ntrcls0	⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)) = ∅) → ((int‘𝐽)‘𝑆) = ∅)

Proof of Theorem ntrcls0

Step	Hyp	Ref	Expression
1		simpl 486	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝐽 ∈ Top)
2		clscld.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	2	clsss3 21774	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
4	2	sscls 21771	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
5	2	ntrss 21770	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑆) ⊆ 𝑋 ∧ 𝑆 ⊆ ((cls‘𝐽)‘𝑆)) → ((int‘𝐽)‘𝑆) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)))
6	1, 3, 4, 5	syl3anc 1369	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)))
7	6	3adant3 1130	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)) = ∅) → ((int‘𝐽)‘𝑆) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)))
8		sseq2 3921	. . . 4 ⊢ (((int‘𝐽)‘((cls‘𝐽)‘𝑆)) = ∅ → (((int‘𝐽)‘𝑆) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ ((int‘𝐽)‘𝑆) ⊆ ∅))
9	8	3ad2ant3 1133	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)) = ∅) → (((int‘𝐽)‘𝑆) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ ((int‘𝐽)‘𝑆) ⊆ ∅))
10	7, 9	mpbid 235	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)) = ∅) → ((int‘𝐽)‘𝑆) ⊆ ∅)
11		ss0 4298	. 2 ⊢ (((int‘𝐽)‘𝑆) ⊆ ∅ → ((int‘𝐽)‘𝑆) = ∅)
12	10, 11	syl 17	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)) = ∅) → ((int‘𝐽)‘𝑆) = ∅)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112   ⊆ wss 3861  ∅c0 4228  ∪ cuni 4802  ‘cfv 6341  Topctop 21608  intcnt 21732  clsccl 21733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5161  ax-sep 5174  ax-nul 5181  ax-pow 5239  ax-pr 5303  ax-un 7466

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4803  df-int 4843  df-iun 4889  df-iin 4890  df-br 5038  df-opab 5100  df-mpt 5118  df-id 5435  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-top 21609  df-cld 21734  df-ntr 21735  df-cls 21736

This theorem is referenced by: (None)