Theorem unopn 20708

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 4450	. . 3 |- ( ( A e. J /\ B e. J ) -> U. { A , B } = ( A u. B ) )
2	1	3adant1 1079	. 2 |- ( ( J e. Top /\ A e. J /\ B e. J ) -> U. { A , B } = ( A u. B ) )
3		prssi 4353	. . . 4 |- ( ( A e. J /\ B e. J ) -> { A , B } C_ J )
4		uniopn 20702	. . . 4 |- ( ( J e. Top /\ { A , B } C_ J ) -> U. { A , B } e. J )
5	3, 4	sylan2 491	. . 3 |- ( ( J e. Top /\ ( A e. J /\ B e. J ) ) -> U. { A , B } e. J )
6	5	3impb 1260	. 2 |- ( ( J e. Top /\ A e. J /\ B e. J ) -> U. { A , B } e. J )
7	2, 6	eqeltrrd 2702	1 |- ( ( J e. Top /\ A e. J /\ B e. J ) -> ( A u. B ) e. J )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   u. cun 3572   C_ wss 3574   {cpr 4179   U.cuni 4436   Topctop 20698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-pw 4160  df-sn 4178  df-pr 4180  df-uni 4437  df-top 20699

This theorem is referenced by:  comppfsc  21335  txcld  21406  icccld  22570  icccncfext  40100