Theorem unopn 22925

Description: The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
unopn	⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∪ 𝐵) ∈ 𝐽)

Proof of Theorem unopn

Step	Hyp	Ref	Expression
1		uniprg 4928	. . 3 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
2	1	3adant1 1129	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
3		prssi 4826	. . . 4 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → {𝐴, 𝐵} ⊆ 𝐽)
4		uniopn 22919	. . . 4 ⊢ ((𝐽 ∈ Top ∧ {𝐴, 𝐵} ⊆ 𝐽) → ∪ {𝐴, 𝐵} ∈ 𝐽)
5	3, 4	sylan2 593	. . 3 ⊢ ((𝐽 ∈ Top ∧ (𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽)) → ∪ {𝐴, 𝐵} ∈ 𝐽)
6	5	3impb 1114	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → ∪ {𝐴, 𝐵} ∈ 𝐽)
7	2, 6	eqeltrrd 2840	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∪ 𝐵) ∈ 𝐽)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ∪ cun 3961   ⊆ wss 3963  {cpr 4633  ∪ cuni 4912  Topctop 22915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-un 3968  df-in 3970  df-ss 3980  df-pw 4607  df-sn 4632  df-pr 4634  df-uni 4913  df-top 22916

This theorem is referenced by:  comppfsc  23556  txcld  23627  icccld  24803  redvmptabs  42369  icccncfext  45843  toplatjoin  48791  topdlat  48793