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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 a topology. (Contributed by Stefan Allan, 27-Feb-2006.) |
Ref | Expression |
---|---|
0opn | ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uni0 4401 | . 2 ⊢ ∪ ∅ = ∅ | |
2 | 0ss 3924 | . . 3 ⊢ ∅ ⊆ 𝐽 | |
3 | uniopn 20527 | . . 3 ⊢ ((𝐽 ∈ Top ∧ ∅ ⊆ 𝐽) → ∪ ∅ ∈ 𝐽) | |
4 | 2, 3 | mpan2 703 | . 2 ⊢ (𝐽 ∈ Top → ∪ ∅ ∈ 𝐽) |
5 | 1, 4 | syl5eqelr 2693 | 1 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ⊆ wss 3540 ∅c0 3874 ∪ cuni 4372 Topctop 20517 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-nul 3875 df-pw 4110 df-sn 4126 df-uni 4373 df-top 20521 |
This theorem is referenced by: 0ntop 20535 topgele 20549 tgclb 20585 0top 20598 en1top 20599 en2top 20600 topcld 20649 clsval2 20664 ntr0 20695 opnnei 20734 0nei 20742 restrcl 20771 rest0 20783 ordtrest2lem 20817 iocpnfordt 20829 icomnfordt 20830 cnindis 20906 iscon2 21027 kqtop 21358 mopn0 22113 locfinref 29236 ordtrest2NEWlem 29296 sxbrsigalem3 29661 cnambfre 32628 |
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