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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 3863 | . 2 ⊢ ∪ ∅ = ∅ | |
2 | 0ss 3486 | . . 3 ⊢ ∅ ⊆ 𝐽 | |
3 | uniopn 14180 | . . 3 ⊢ ((𝐽 ∈ Top ∧ ∅ ⊆ 𝐽) → ∪ ∅ ∈ 𝐽) | |
4 | 2, 3 | mpan2 425 | . 2 ⊢ (𝐽 ∈ Top → ∪ ∅ ∈ 𝐽) |
5 | 1, 4 | eqeltrrid 2281 | 1 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2164 ⊆ wss 3154 ∅c0 3447 ∪ cuni 3836 Topctop 14176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2175 ax-sep 4148 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-dif 3156 df-in 3160 df-ss 3167 df-nul 3448 df-pw 3604 df-sn 3625 df-uni 3837 df-top 14177 |
This theorem is referenced by: 0ntop 14186 topgele 14208 istps 14211 topontopn 14216 tgclb 14244 en1top 14256 topcld 14288 ntr0 14313 0nei 14345 restrcl 14346 rest0 14358 mopn0 14667 |
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