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Theorem unopn 21115
Description: The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
unopn ((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)

Proof of Theorem unopn
StepHypRef Expression
1 uniprg 4685 . . 3 ((𝐴𝐽𝐵𝐽) → {𝐴, 𝐵} = (𝐴𝐵))
213adant1 1121 . 2 ((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → {𝐴, 𝐵} = (𝐴𝐵))
3 prssi 4583 . . . 4 ((𝐴𝐽𝐵𝐽) → {𝐴, 𝐵} ⊆ 𝐽)
4 uniopn 21109 . . . 4 ((𝐽 ∈ Top ∧ {𝐴, 𝐵} ⊆ 𝐽) → {𝐴, 𝐵} ∈ 𝐽)
53, 4sylan2 586 . . 3 ((𝐽 ∈ Top ∧ (𝐴𝐽𝐵𝐽)) → {𝐴, 𝐵} ∈ 𝐽)
653impb 1104 . 2 ((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → {𝐴, 𝐵} ∈ 𝐽)
72, 6eqeltrrd 2859 1 ((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2106  cun 3789  wss 3791  {cpr 4399   cuni 4671  Topctop 21105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753  ax-sep 5017
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-v 3399  df-un 3796  df-in 3798  df-ss 3805  df-pw 4380  df-sn 4398  df-pr 4400  df-uni 4672  df-top 21106
This theorem is referenced by:  comppfsc  21744  txcld  21815  icccld  22978  icccncfext  41010
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