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Mirrors > Home > MPE Home > Th. List > 0opn | Structured version Visualization version GIF version |
Description: The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.) |
Ref | Expression |
---|---|
0opn | ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uni0 4866 | . 2 ⊢ ∪ ∅ = ∅ | |
2 | 0ss 4350 | . . 3 ⊢ ∅ ⊆ 𝐽 | |
3 | uniopn 21505 | . . 3 ⊢ ((𝐽 ∈ Top ∧ ∅ ⊆ 𝐽) → ∪ ∅ ∈ 𝐽) | |
4 | 2, 3 | mpan2 689 | . 2 ⊢ (𝐽 ∈ Top → ∪ ∅ ∈ 𝐽) |
5 | 1, 4 | eqeltrrid 2918 | 1 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ⊆ wss 3936 ∅c0 4291 ∪ cuni 4838 Topctop 21501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 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-dif 3939 df-in 3943 df-ss 3952 df-nul 4292 df-pw 4541 df-sn 4568 df-uni 4839 df-top 21502 |
This theorem is referenced by: 0ntop 21513 topgele 21538 tgclb 21578 0top 21591 en1top 21592 en2top 21593 topcld 21643 clsval2 21658 ntr0 21689 opnnei 21728 0nei 21736 restrcl 21765 rest0 21777 ordtrest2lem 21811 iocpnfordt 21823 icomnfordt 21824 cnindis 21900 isconn2 22022 kqtop 22353 mopn0 23108 locfinref 31105 ordtrest2NEWlem 31165 sxbrsigalem3 31530 cnambfre 34955 |
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