Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 3904 | . . 3 ⊢ (∪ 𝐽 ∖ ∅) = ∪ 𝐽 | |
2 | 1 | topopn 20536 | . 2 ⊢ (𝐽 ∈ Top → (∪ 𝐽 ∖ ∅) ∈ 𝐽) |
3 | 0ss 3924 | . . 3 ⊢ ∅ ⊆ ∪ 𝐽 | |
4 | eqid 2610 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
5 | 4 | iscld2 20642 | . . 3 ⊢ ((𝐽 ∈ Top ∧ ∅ ⊆ ∪ 𝐽) → (∅ ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ ∅) ∈ 𝐽)) |
6 | 3, 5 | mpan2 703 | . 2 ⊢ (𝐽 ∈ Top → (∅ ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ ∅) ∈ 𝐽)) |
7 | 2, 6 | mpbird 246 | 1 ⊢ (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∈ wcel 1977 ∖ cdif 3537 ⊆ wss 3540 ∅c0 3874 ∪ cuni 4372 ‘cfv 5804 Topctop 20517 Clsdccld 20630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-top 20521 df-cld 20633 |
This theorem is referenced by: cls0 20694 indiscld 20705 iscldtop 20709 iccordt 20828 iscon2 21027 tgptsmscld 21764 mblfinlem2 32617 mblfinlem3 32618 ismblfin 32620 |
Copyright terms: Public domain | W3C validator |