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Mirrors > Home > MPE Home > Th. List > topcld | Structured version Visualization version GIF version |
Description: The underlying set of a topology is closed. Part of Theorem 6.1(1) of [Munkres] p. 93. (Contributed by NM, 3-Oct-2006.) |
Ref | Expression |
---|---|
iscld.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
topcld | ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difid 4179 | . . . 4 ⊢ (𝑋 ∖ 𝑋) = ∅ | |
2 | 0opn 21080 | . . . 4 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | |
3 | 1, 2 | syl5eqel 2911 | . . 3 ⊢ (𝐽 ∈ Top → (𝑋 ∖ 𝑋) ∈ 𝐽) |
4 | ssid 3849 | . . 3 ⊢ 𝑋 ⊆ 𝑋 | |
5 | 3, 4 | jctil 517 | . 2 ⊢ (𝐽 ∈ Top → (𝑋 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑋) ∈ 𝐽)) |
6 | iscld.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
7 | 6 | iscld 21203 | . 2 ⊢ (𝐽 ∈ Top → (𝑋 ∈ (Clsd‘𝐽) ↔ (𝑋 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑋) ∈ 𝐽))) |
8 | 5, 7 | mpbird 249 | 1 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∖ cdif 3796 ⊆ wss 3799 ∅c0 4145 ∪ cuni 4659 ‘cfv 6124 Topctop 21069 Clsdccld 21192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-iota 6087 df-fun 6126 df-fv 6132 df-top 21070 df-cld 21195 |
This theorem is referenced by: clsval 21213 riincld 21220 clscld 21223 clstop 21245 cldmre 21254 indiscld 21267 isconn2 21589 cnmpt2pc 23098 rlmbn 23530 ubthlem1 28282 unicls 30495 cmpfiiin 38105 kelac1 38477 |
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