Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nrmsep2 | Structured version Visualization version GIF version |
Description: In a normal space, any two disjoint closed sets have the property that each one is a subset of an open set whose closure is disjoint from the other. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 24-Aug-2015.) |
Ref | Expression |
---|---|
nrmsep2 | ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 486 | . . 3 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → 𝐽 ∈ Nrm) | |
2 | simpr2 1196 | . . . 4 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → 𝐷 ∈ (Clsd‘𝐽)) | |
3 | eqid 2738 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
4 | 3 | cldopn 21775 | . . . 4 ⊢ (𝐷 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝐷) ∈ 𝐽) |
5 | 2, 4 | syl 17 | . . 3 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → (∪ 𝐽 ∖ 𝐷) ∈ 𝐽) |
6 | simpr1 1195 | . . 3 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → 𝐶 ∈ (Clsd‘𝐽)) | |
7 | simpr3 1197 | . . . 4 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → (𝐶 ∩ 𝐷) = ∅) | |
8 | 3 | cldss 21773 | . . . . 5 ⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐶 ⊆ ∪ 𝐽) |
9 | reldisj 4338 | . . . . 5 ⊢ (𝐶 ⊆ ∪ 𝐽 → ((𝐶 ∩ 𝐷) = ∅ ↔ 𝐶 ⊆ (∪ 𝐽 ∖ 𝐷))) | |
10 | 6, 8, 9 | 3syl 18 | . . . 4 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → ((𝐶 ∩ 𝐷) = ∅ ↔ 𝐶 ⊆ (∪ 𝐽 ∖ 𝐷))) |
11 | 7, 10 | mpbid 235 | . . 3 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → 𝐶 ⊆ (∪ 𝐽 ∖ 𝐷)) |
12 | nrmsep3 22099 | . . 3 ⊢ ((𝐽 ∈ Nrm ∧ ((∪ 𝐽 ∖ 𝐷) ∈ 𝐽 ∧ 𝐶 ∈ (Clsd‘𝐽) ∧ 𝐶 ⊆ (∪ 𝐽 ∖ 𝐷))) → ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ (∪ 𝐽 ∖ 𝐷))) | |
13 | 1, 5, 6, 11, 12 | syl13anc 1373 | . 2 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ (∪ 𝐽 ∖ 𝐷))) |
14 | ssdifin0 4369 | . . . 4 ⊢ (((cls‘𝐽)‘𝑥) ⊆ (∪ 𝐽 ∖ 𝐷) → (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅) | |
15 | 14 | anim2i 620 | . . 3 ⊢ ((𝐶 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ (∪ 𝐽 ∖ 𝐷)) → (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅)) |
16 | 15 | reximi 3156 | . 2 ⊢ (∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ (∪ 𝐽 ∖ 𝐷)) → ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅)) |
17 | 13, 16 | syl 17 | 1 ⊢ ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶 ∩ 𝐷) = ∅)) → ∃𝑥 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2113 ∃wrex 3054 ∖ cdif 3838 ∩ cin 3840 ⊆ wss 3841 ∅c0 4209 ∪ cuni 4793 ‘cfv 6333 Clsdccld 21760 clsccl 21762 Nrmcnrm 22054 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6291 df-fun 6335 df-fn 6336 df-fv 6341 df-top 21638 df-cld 21763 df-nrm 22061 |
This theorem is referenced by: nrmsep 22101 isnrm2 22102 |
Copyright terms: Public domain | W3C validator |