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Theorem unopn 16978
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 4032 . . 3  |-  ( ( A  e.  J  /\  B  e.  J )  ->  U. { A ,  B }  =  ( A  u.  B )
)
213adant1 976 . 2  |-  ( ( J  e.  Top  /\  A  e.  J  /\  B  e.  J )  ->  U. { A ,  B }  =  ( A  u.  B )
)
3 prssi 3956 . . . 4  |-  ( ( A  e.  J  /\  B  e.  J )  ->  { A ,  B }  C_  J )
4 uniopn 16972 . . . 4  |-  ( ( J  e.  Top  /\  { A ,  B }  C_  J )  ->  U. { A ,  B }  e.  J )
53, 4sylan2 462 . . 3  |-  ( ( J  e.  Top  /\  ( A  e.  J  /\  B  e.  J
) )  ->  U. { A ,  B }  e.  J )
653impb 1150 . 2  |-  ( ( J  e.  Top  /\  A  e.  J  /\  B  e.  J )  ->  U. { A ,  B }  e.  J
)
72, 6eqeltrrd 2513 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726    u. cun 3320    C_ wss 3322   {cpr 3817   U.cuni 4017   Topctop 16960
This theorem is referenced by:  txcld  17637  icccld  18803  comppfsc  26389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-v 2960  df-un 3327  df-in 3329  df-ss 3336  df-pw 3803  df-sn 3822  df-pr 3823  df-uni 4018  df-top 16965
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