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Mirrors > Home > MPE Home > Th. List > 0cld | Structured version Visualization version GIF version |
Description: The empty set is closed. Part of Theorem 6.1(1) of [Munkres] p. 93. (Contributed by NM, 4-Oct-2006.) |
Ref | Expression |
---|---|
0cld | ⊢ (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dif0 4329 | . . 3 ⊢ (∪ 𝐽 ∖ ∅) = ∪ 𝐽 | |
2 | 1 | topopn 21442 | . 2 ⊢ (𝐽 ∈ Top → (∪ 𝐽 ∖ ∅) ∈ 𝐽) |
3 | 0ss 4347 | . . 3 ⊢ ∅ ⊆ ∪ 𝐽 | |
4 | eqid 2818 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
5 | 4 | iscld2 21564 | . . 3 ⊢ ((𝐽 ∈ Top ∧ ∅ ⊆ ∪ 𝐽) → (∅ ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ ∅) ∈ 𝐽)) |
6 | 3, 5 | mpan2 687 | . 2 ⊢ (𝐽 ∈ Top → (∅ ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ ∅) ∈ 𝐽)) |
7 | 2, 6 | mpbird 258 | 1 ⊢ (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∈ wcel 2105 ∖ cdif 3930 ⊆ wss 3933 ∅c0 4288 ∪ cuni 4830 ‘cfv 6348 Topctop 21429 Clsdccld 21552 |
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 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
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 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-top 21430 df-cld 21555 |
This theorem is referenced by: cls0 21616 indiscld 21627 iscldtop 21631 iccordt 21750 isconn2 21950 tgptsmscld 22686 mblfinlem2 34811 mblfinlem3 34812 ismblfin 34814 |
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