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Mirrors > Home > ILE Home > Th. List > ntrcls0 | GIF version |
Description: A subset whose closure has an empty interior also has an empty interior. (Contributed by NM, 4-Oct-2007.) |
Ref | Expression |
---|---|
clscld.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
ntrcls0 | ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)) = ∅) → ((int‘𝐽)‘𝑆) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝐽 ∈ Top) | |
2 | clscld.1 | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | clsss3 12677 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋) |
4 | 2 | sscls 12667 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆)) |
5 | 2 | ntrss 12666 | . . . . 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 1227 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑆))) |
7 | 6 | 3adant3 1006 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)) = ∅) → ((int‘𝐽)‘𝑆) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑆))) |
8 | sseq2 3161 | . . . 4 ⊢ (((int‘𝐽)‘((cls‘𝐽)‘𝑆)) = ∅ → (((int‘𝐽)‘𝑆) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ ((int‘𝐽)‘𝑆) ⊆ ∅)) | |
9 | 8 | 3ad2ant3 1009 | . . 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 3444 | . 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) = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 967 = wceq 1342 ∈ wcel 2135 ⊆ wss 3111 ∅c0 3404 ∪ cuni 3783 ‘cfv 5182 Topctop 12542 intcnt 12640 clsccl 12641 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-top 12543 df-cld 12642 df-ntr 12643 df-cls 12644 |
This theorem is referenced by: (None) |
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