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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 21506 | . . . 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 522 | . 2 ⊢ (𝐽 ∈ Top → (𝑋 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑋) ∈ 𝐽)) |
6 | iscld.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
7 | 6 | iscld 21629 | . 2 ⊢ (𝐽 ∈ Top → (𝑋 ∈ (Clsd‘𝐽) ↔ (𝑋 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑋) ∈ 𝐽))) |
8 | 5, 7 | mpbird 259 | 1 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∖ cdif 3932 ⊆ wss 3935 ∅c0 4290 ∪ cuni 4831 ‘cfv 6349 Topctop 21495 Clsdccld 21618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 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 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 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 21496 df-cld 21621 |
This theorem is referenced by: clsval 21639 riincld 21646 clscld 21649 clstop 21671 cldmre 21680 indiscld 21693 isconn2 22016 cnmpopc 23526 rlmbn 23958 ubthlem1 28641 unicls 31141 cmpfiiin 39287 kelac1 39656 |
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