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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 3851 | . 2 ⊢ ∪ ∅ = ∅ | |
2 | 0ss 3476 | . . 3 ⊢ ∅ ⊆ 𝐽 | |
3 | uniopn 13978 | . . 3 ⊢ ((𝐽 ∈ Top ∧ ∅ ⊆ 𝐽) → ∪ ∅ ∈ 𝐽) | |
4 | 2, 3 | mpan2 425 | . 2 ⊢ (𝐽 ∈ Top → ∪ ∅ ∈ 𝐽) |
5 | 1, 4 | eqeltrrid 2277 | 1 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2160 ⊆ wss 3144 ∅c0 3437 ∪ cuni 3824 Topctop 13974 |
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 2171 ax-sep 4136 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-uni 3825 df-top 13975 |
This theorem is referenced by: 0ntop 13984 topgele 14006 istps 14009 topontopn 14014 tgclb 14042 en1top 14054 topcld 14086 ntr0 14111 0nei 14143 restrcl 14144 rest0 14156 mopn0 14465 |
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