Theorem ntrcls0 12925

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 108	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝐽 ∈ Top)
2		clscld.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	2	clsss3 12924	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋)
4	2	sscls 12914	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
5	2	ntrss 12913	. . . . 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 1233	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)))
7	6	3adant3 1012	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)) = ∅) → ((int‘𝐽)‘𝑆) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)))
8		sseq2 3171	. . . 4 ⊢ (((int‘𝐽)‘((cls‘𝐽)‘𝑆)) = ∅ → (((int‘𝐽)‘𝑆) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ ((int‘𝐽)‘𝑆) ⊆ ∅))
9	8	3ad2ant3 1015	. . 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 146	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)) = ∅) → ((int‘𝐽)‘𝑆) ⊆ ∅)
11		ss0 3455	. 2 ⊢ (((int‘𝐽)‘𝑆) ⊆ ∅ → ((int‘𝐽)‘𝑆) = ∅)
12	10, 11	syl 14	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)) = ∅) → ((int‘𝐽)‘𝑆) = ∅)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 973   = wceq 1348   ∈ wcel 2141   ⊆ wss 3121  ∅c0 3414  ∪ cuni 3796  ‘cfv 5198  Topctop 12789  intcnt 12887  clsccl 12888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-top 12790  df-cld 12889  df-ntr 12890  df-cls 12891

This theorem is referenced by: (None)