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Mirrors > Home > ILE Home > Th. List > unopn | GIF version |
Description: The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
unopn | ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∪ 𝐵) ∈ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniprg 3809 | . . 3 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) | |
2 | 1 | 3adant1 1010 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) |
3 | prssi 3736 | . . . 4 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → {𝐴, 𝐵} ⊆ 𝐽) | |
4 | uniopn 12714 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ {𝐴, 𝐵} ⊆ 𝐽) → ∪ {𝐴, 𝐵} ∈ 𝐽) | |
5 | 3, 4 | sylan2 284 | . . 3 ⊢ ((𝐽 ∈ Top ∧ (𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽)) → ∪ {𝐴, 𝐵} ∈ 𝐽) |
6 | 5 | 3impb 1194 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → ∪ {𝐴, 𝐵} ∈ 𝐽) |
7 | 2, 6 | eqeltrrd 2248 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∪ 𝐵) ∈ 𝐽) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 973 = wceq 1348 ∈ wcel 2141 ∪ cun 3119 ⊆ wss 3121 {cpr 3582 ∪ cuni 3794 Topctop 12710 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-sep 4105 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-uni 3795 df-top 12711 |
This theorem is referenced by: reopnap 13253 |
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