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Theorem unopn 12718
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 3809 . . 3 ((𝐴𝐽𝐵𝐽) → {𝐴, 𝐵} = (𝐴𝐵))
213adant1 1010 . 2 ((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → {𝐴, 𝐵} = (𝐴𝐵))
3 prssi 3736 . . . 4 ((𝐴𝐽𝐵𝐽) → {𝐴, 𝐵} ⊆ 𝐽)
4 uniopn 12714 . . . 4 ((𝐽 ∈ Top ∧ {𝐴, 𝐵} ⊆ 𝐽) → {𝐴, 𝐵} ∈ 𝐽)
53, 4sylan2 284 . . 3 ((𝐽 ∈ Top ∧ (𝐴𝐽𝐵𝐽)) → {𝐴, 𝐵} ∈ 𝐽)
653impb 1194 . 2 ((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → {𝐴, 𝐵} ∈ 𝐽)
72, 6eqeltrrd 2248 1 ((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 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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