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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 4329 | . . . 4 ⊢ (𝑋 ∖ 𝑋) = ∅ | |
2 | 0opn 21442 | . . . 4 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | |
3 | 1, 2 | eqeltrid 2917 | . . 3 ⊢ (𝐽 ∈ Top → (𝑋 ∖ 𝑋) ∈ 𝐽) |
4 | ssid 3988 | . . 3 ⊢ 𝑋 ⊆ 𝑋 | |
5 | 3, 4 | jctil 520 | . 2 ⊢ (𝐽 ∈ Top → (𝑋 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑋) ∈ 𝐽)) |
6 | iscld.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
7 | 6 | iscld 21565 | . 2 ⊢ (𝐽 ∈ Top → (𝑋 ∈ (Clsd‘𝐽) ↔ (𝑋 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑋) ∈ 𝐽))) |
8 | 5, 7 | mpbird 258 | 1 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∖ cdif 3932 ⊆ wss 3935 ∅c0 4290 ∪ cuni 4832 ‘cfv 6349 Topctop 21431 Clsdccld 21554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-top 21432 df-cld 21557 |
This theorem is referenced by: clsval 21575 riincld 21582 clscld 21585 clstop 21607 cldmre 21616 indiscld 21629 isconn2 21952 cnmpopc 23461 rlmbn 23893 ubthlem1 28575 unicls 31046 cmpfiiin 39174 kelac1 39543 |
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