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Theorem unopn 11872
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 3690 . . 3 ((𝐴𝐽𝐵𝐽) → {𝐴, 𝐵} = (𝐴𝐵))
213adant1 964 . 2 ((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → {𝐴, 𝐵} = (𝐴𝐵))
3 prssi 3617 . . . 4 ((𝐴𝐽𝐵𝐽) → {𝐴, 𝐵} ⊆ 𝐽)
4 uniopn 11868 . . . 4 ((𝐽 ∈ Top ∧ {𝐴, 𝐵} ⊆ 𝐽) → {𝐴, 𝐵} ∈ 𝐽)
53, 4sylan2 281 . . 3 ((𝐽 ∈ Top ∧ (𝐴𝐽𝐵𝐽)) → {𝐴, 𝐵} ∈ 𝐽)
653impb 1142 . 2 ((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → {𝐴, 𝐵} ∈ 𝐽)
72, 6eqeltrrd 2172 1 ((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 927   = wceq 1296  wcel 1445  cun 3011  wss 3013  {cpr 3467   cuni 3675  Topctop 11864
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-uni 3676  df-top 11865
This theorem is referenced by: (None)
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