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Theorem unopn 12188
 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 3751 . . 3 ((𝐴𝐽𝐵𝐽) → {𝐴, 𝐵} = (𝐴𝐵))
213adant1 999 . 2 ((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → {𝐴, 𝐵} = (𝐴𝐵))
3 prssi 3678 . . . 4 ((𝐴𝐽𝐵𝐽) → {𝐴, 𝐵} ⊆ 𝐽)
4 uniopn 12184 . . . 4 ((𝐽 ∈ Top ∧ {𝐴, 𝐵} ⊆ 𝐽) → {𝐴, 𝐵} ∈ 𝐽)
53, 4sylan2 284 . . 3 ((𝐽 ∈ Top ∧ (𝐴𝐽𝐵𝐽)) → {𝐴, 𝐵} ∈ 𝐽)
653impb 1177 . 2 ((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → {𝐴, 𝐵} ∈ 𝐽)
72, 6eqeltrrd 2217 1 ((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 962   = wceq 1331   ∈ wcel 1480   ∪ cun 3069   ⊆ wss 3071  {cpr 3528  ∪ cuni 3736  Topctop 12180 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3737  df-top 12181 This theorem is referenced by:  reopnap  12723
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