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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 3190 | . . 3 ⊢ 𝐽 ⊆ 𝐽 | |
3 | uniopn 13961 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐽 ⊆ 𝐽) → ∪ 𝐽 ∈ 𝐽) | |
4 | 2, 3 | mpan2 425 | . 2 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽) |
5 | 1, 4 | eqeltrid 2276 | 1 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 ⊆ wss 3144 ∪ cuni 3824 Topctop 13957 |
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-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-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-in 3150 df-ss 3157 df-pw 3592 df-uni 3825 df-top 13958 |
This theorem is referenced by: toponmax 13985 cldval 14059 ntrfval 14060 clsfval 14061 iscld 14063 ntrval 14070 clsval 14071 0cld 14072 ntrtop 14088 neifval 14100 neif 14101 neival 14103 isnei 14104 tpnei 14120 cnrest 14195 txcn 14235 |
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