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Mirrors > Home > ILE Home > Th. List > 0opn | 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 3702 | . 2 ⊢ ∪ ∅ = ∅ | |
2 | 0ss 3340 | . . 3 ⊢ ∅ ⊆ 𝐽 | |
3 | uniopn 11868 | . . 3 ⊢ ((𝐽 ∈ Top ∧ ∅ ⊆ 𝐽) → ∪ ∅ ∈ 𝐽) | |
4 | 2, 3 | mpan2 417 | . 2 ⊢ (𝐽 ∈ Top → ∪ ∅ ∈ 𝐽) |
5 | 1, 4 | syl5eqelr 2182 | 1 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1445 ⊆ wss 3013 ∅c0 3302 ∪ cuni 3675 Topctop 11864 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-dif 3015 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-uni 3676 df-top 11865 |
This theorem is referenced by: 0ntop 11874 topgele 11895 istps 11898 topontopn 11903 tgclb 11933 en1top 11945 topcld 11977 ntr0 12002 0nei 12034 restrcl 12035 rest0 12047 mopn0 12290 |
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