Theorem ntrcls0 23100

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 482	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝐽 ∈ Top)
2		clscld.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	2	clsss3 23083	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
4	2	sscls 23080	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
5	2	ntrss 23079	. . . . 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 1370	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)))
7	6	3adant3 1131	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)) = ∅) → ((int‘𝐽)‘𝑆) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)))
8		sseq2 4022	. . . 4 ⊢ (((int‘𝐽)‘((cls‘𝐽)‘𝑆)) = ∅ → (((int‘𝐽)‘𝑆) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ ((int‘𝐽)‘𝑆) ⊆ ∅))
9	8	3ad2ant3 1134	. . 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 232	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)) = ∅) → ((int‘𝐽)‘𝑆) ⊆ ∅)
11		ss0 4408	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ⊆ wss 3963  ∅c0 4339  ∪ cuni 4912  ‘cfv 6563  Topctop 22915  intcnt 23041  clsccl 23042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-top 22916  df-cld 23043  df-ntr 23044  df-cls 23045

This theorem is referenced by: (None)