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Theorem unopn 22804
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 4924 . . 3 ((𝐴𝐽𝐵𝐽) → {𝐴, 𝐵} = (𝐴𝐵))
213adant1 1128 . 2 ((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → {𝐴, 𝐵} = (𝐴𝐵))
3 prssi 4825 . . . 4 ((𝐴𝐽𝐵𝐽) → {𝐴, 𝐵} ⊆ 𝐽)
4 uniopn 22798 . . . 4 ((𝐽 ∈ Top ∧ {𝐴, 𝐵} ⊆ 𝐽) → {𝐴, 𝐵} ∈ 𝐽)
53, 4sylan2 592 . . 3 ((𝐽 ∈ Top ∧ (𝐴𝐽𝐵𝐽)) → {𝐴, 𝐵} ∈ 𝐽)
653impb 1113 . 2 ((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → {𝐴, 𝐵} ∈ 𝐽)
72, 6eqeltrrd 2830 1 ((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 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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