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Mirrors > Home > MPE Home > Th. List > Mathboxes > sepcsepo | Structured version Visualization version GIF version |
Description: If two sets are separated by closed neighborhoods, then they are separated by (open) neighborhoods. See sepnsepo 45656 for the equivalence betewen separatedness by open neighborhoods and separatedness by neighborhoods. Although 𝐽 ∈ Top might be redundant here, it is listed for explicitness. (Contributed by Zhi Wang, 8-Sep-2024.) |
Ref | Expression |
---|---|
sepdisj.1 | ⊢ (𝜑 → 𝐽 ∈ Top) |
sepcsepo.2 | ⊢ (𝜑 → ∃𝑛 ∈ ((nei‘𝐽)‘𝑆)∃𝑚 ∈ ((nei‘𝐽)‘𝑇)(𝑛 ∈ (Clsd‘𝐽) ∧ 𝑚 ∈ (Clsd‘𝐽) ∧ (𝑛 ∩ 𝑚) = ∅)) |
Ref | Expression |
---|---|
sepcsepo | ⊢ (𝜑 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sepcsepo.2 | . . 3 ⊢ (𝜑 → ∃𝑛 ∈ ((nei‘𝐽)‘𝑆)∃𝑚 ∈ ((nei‘𝐽)‘𝑇)(𝑛 ∈ (Clsd‘𝐽) ∧ 𝑚 ∈ (Clsd‘𝐽) ∧ (𝑛 ∩ 𝑚) = ∅)) | |
2 | simp3 1135 | . . . . 5 ⊢ ((𝑛 ∈ (Clsd‘𝐽) ∧ 𝑚 ∈ (Clsd‘𝐽) ∧ (𝑛 ∩ 𝑚) = ∅) → (𝑛 ∩ 𝑚) = ∅) | |
3 | 2 | reximi 3171 | . . . 4 ⊢ (∃𝑚 ∈ ((nei‘𝐽)‘𝑇)(𝑛 ∈ (Clsd‘𝐽) ∧ 𝑚 ∈ (Clsd‘𝐽) ∧ (𝑛 ∩ 𝑚) = ∅) → ∃𝑚 ∈ ((nei‘𝐽)‘𝑇)(𝑛 ∩ 𝑚) = ∅) |
4 | 3 | reximi 3171 | . . 3 ⊢ (∃𝑛 ∈ ((nei‘𝐽)‘𝑆)∃𝑚 ∈ ((nei‘𝐽)‘𝑇)(𝑛 ∈ (Clsd‘𝐽) ∧ 𝑚 ∈ (Clsd‘𝐽) ∧ (𝑛 ∩ 𝑚) = ∅) → ∃𝑛 ∈ ((nei‘𝐽)‘𝑆)∃𝑚 ∈ ((nei‘𝐽)‘𝑇)(𝑛 ∩ 𝑚) = ∅) |
5 | 1, 4 | syl 17 | . 2 ⊢ (𝜑 → ∃𝑛 ∈ ((nei‘𝐽)‘𝑆)∃𝑚 ∈ ((nei‘𝐽)‘𝑇)(𝑛 ∩ 𝑚) = ∅) |
6 | sepdisj.1 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Top) | |
7 | 6 | sepnsepo 45656 | . 2 ⊢ (𝜑 → (∃𝑛 ∈ ((nei‘𝐽)‘𝑆)∃𝑚 ∈ ((nei‘𝐽)‘𝑇)(𝑛 ∩ 𝑚) = ∅ ↔ ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))) |
8 | 5, 7 | mpbid 235 | 1 ⊢ (𝜑 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∃wrex 3071 ∩ cin 3859 ⊆ wss 3860 ∅c0 4227 ‘cfv 6340 Topctop 21606 Clsdccld 21729 neicnei 21810 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-ral 3075 df-rex 3076 df-reu 3077 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4802 df-iun 4888 df-br 5037 df-opab 5099 df-mpt 5117 df-id 5434 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-top 21607 df-nei 21811 |
This theorem is referenced by: (None) |
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