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Mirrors > Home > ILE Home > Th. List > topopn | GIF version |
Description: The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.) |
Ref | Expression |
---|---|
1open.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
topopn | ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1open.1 | . 2 ⊢ 𝑋 = ∪ 𝐽 | |
2 | ssid 3059 | . . 3 ⊢ 𝐽 ⊆ 𝐽 | |
3 | uniopn 11867 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐽 ⊆ 𝐽) → ∪ 𝐽 ∈ 𝐽) | |
4 | 2, 3 | mpan2 417 | . 2 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽) |
5 | 1, 4 | syl5eqel 2181 | 1 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1296 ∈ wcel 1445 ⊆ wss 3013 ∪ cuni 3675 Topctop 11863 |
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-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-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-in 3019 df-ss 3026 df-pw 3451 df-uni 3676 df-top 11864 |
This theorem is referenced by: toponmax 11890 cldval 11966 ntrfval 11967 clsfval 11968 iscld 11970 ntrval 11977 clsval 11978 0cld 11979 ntrtop 11995 neifval 12007 neif 12008 neival 12010 isnei 12011 tpnei 12027 cnrest 12101 |
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