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Mirrors > Home > MPE Home > Th. List > unopn | Structured version Visualization version 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 4924 | . . 3 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) | |
2 | 1 | 3adant1 1128 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) |
3 | prssi 4825 | . . . 4 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → {𝐴, 𝐵} ⊆ 𝐽) | |
4 | uniopn 22798 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ {𝐴, 𝐵} ⊆ 𝐽) → ∪ {𝐴, 𝐵} ∈ 𝐽) | |
5 | 3, 4 | sylan2 592 | . . 3 ⊢ ((𝐽 ∈ Top ∧ (𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽)) → ∪ {𝐴, 𝐵} ∈ 𝐽) |
6 | 5 | 3impb 1113 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → ∪ {𝐴, 𝐵} ∈ 𝐽) |
7 | 2, 6 | eqeltrrd 2830 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∪ 𝐵) ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∪ cun 3945 ⊆ wss 3947 {cpr 4631 ∪ cuni 4908 Topctop 22794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-un 3952 df-in 3954 df-ss 3964 df-pw 4605 df-sn 4630 df-pr 4632 df-uni 4909 df-top 22795 |
This theorem is referenced by: comppfsc 23435 txcld 23506 icccld 24682 icccncfext 45275 toplatjoin 48013 topdlat 48015 |
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