Theorem unopn 22847

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 4879	. . 3 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
2	1	3adant1 1130	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
3		prssi 4777	. . . 4 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → {𝐴, 𝐵} ⊆ 𝐽)
4		uniopn 22841	. . . 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 2837	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∪ 𝐵) ∈ 𝐽)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ∪ cun 3899   ⊆ wss 3901  {cpr 4582  ∪ cuni 4863  Topctop 22837

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-un 3906  df-in 3908  df-ss 3918  df-pw 4556  df-sn 4581  df-pr 4583  df-uni 4864  df-top 22838

This theorem is referenced by:  comppfsc  23476  txcld  23547  icccld  24710  redvmptabs  42611  icccncfext  46127  toplatjoin  49243  topdlat  49245