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Theorem unopn 16649
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
StepHypRef Expression
1 uniprg 3842 . . 3  |-  ( ( A  e.  J  /\  B  e.  J )  ->  U. { A ,  B }  =  ( A  u.  B )
)
213adant1 973 . 2  |-  ( ( J  e.  Top  /\  A  e.  J  /\  B  e.  J )  ->  U. { A ,  B }  =  ( A  u.  B )
)
3 prssi 3771 . . . 4  |-  ( ( A  e.  J  /\  B  e.  J )  ->  { A ,  B }  C_  J )
4 uniopn 16643 . . . 4  |-  ( ( J  e.  Top  /\  { A ,  B }  C_  J )  ->  U. { A ,  B }  e.  J )
53, 4sylan2 460 . . 3  |-  ( ( J  e.  Top  /\  ( A  e.  J  /\  B  e.  J
) )  ->  U. { A ,  B }  e.  J )
653impb 1147 . 2  |-  ( ( J  e.  Top  /\  A  e.  J  /\  B  e.  J )  ->  U. { A ,  B }  e.  J
)
72, 6eqeltrrd 2358 1  |-  ( ( J  e.  Top  /\  A  e.  J  /\  B  e.  J )  ->  ( A  u.  B
)  e.  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    u. cun 3150    C_ wss 3152   {cpr 3641   U.cuni 3827   Topctop 16631
This theorem is referenced by:  txcld  17298  icccld  18276  comppfsc  26307  unopnOLD  26464
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-un 3157  df-in 3159  df-ss 3166  df-pw 3627  df-sn 3646  df-pr 3647  df-uni 3828  df-top 16636
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