Theorem unopn 18658

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

Assertion

Ref	Expression
unopn	|- ( ( J e. Top /\ A e. J /\ B e. J ) -> ( A u. B ) e. J )

Proof of Theorem unopn

Step	Hyp	Ref	Expression
1		uniprg 4216	. . 3 |- ( ( A e. J /\ B e. J ) -> U. { A , B } = ( A u. B ) )
2	1	3adant1 1006	. 2 |- ( ( J e. Top /\ A e. J /\ B e. J ) -> U. { A , B } = ( A u. B ) )
3		prssi 4140	. . . 4 |- ( ( A e. J /\ B e. J ) -> { A , B } C_ J )
4		uniopn 18652	. . . 4 |- ( ( J e. Top /\ { A , B } C_ J ) -> U. { A , B } e. J )
5	3, 4	sylan2 474	. . 3 |- ( ( J e. Top /\ ( A e. J /\ B e. J ) ) -> U. { A , B } e. J )
6	5	3impb 1184	. 2 |- ( ( J e. Top /\ A e. J /\ B e. J ) -> U. { A , B } e. J )
7	2, 6	eqeltrrd 2543	1 |- ( ( J e. Top /\ A e. J /\ B e. J ) -> ( A u. B ) e. J )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   u. cun 3437   C_ wss 3439   {cpr 3990   U.cuni 4202   Topctop 18640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-v 3080  df-un 3444  df-in 3446  df-ss 3453  df-pw 3973  df-sn 3989  df-pr 3991  df-uni 4203  df-top 18645

This theorem is referenced by:  txcld  19318  icccld  20488  comppfsc  28750