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Theorem unopn 19596
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 4204 . . 3  |-  ( ( A  e.  J  /\  B  e.  J )  ->  U. { A ,  B }  =  ( A  u.  B )
)
213adant1 1015 . 2  |-  ( ( J  e.  Top  /\  A  e.  J  /\  B  e.  J )  ->  U. { A ,  B }  =  ( A  u.  B )
)
3 prssi 4127 . . . 4  |-  ( ( A  e.  J  /\  B  e.  J )  ->  { A ,  B }  C_  J )
4 uniopn 19590 . . . 4  |-  ( ( J  e.  Top  /\  { A ,  B }  C_  J )  ->  U. { A ,  B }  e.  J )
53, 4sylan2 472 . . 3  |-  ( ( J  e.  Top  /\  ( A  e.  J  /\  B  e.  J
) )  ->  U. { A ,  B }  e.  J )
653impb 1193 . 2  |-  ( ( J  e.  Top  /\  A  e.  J  /\  B  e.  J )  ->  U. { A ,  B }  e.  J
)
72, 6eqeltrrd 2491 1  |-  ( ( J  e.  Top  /\  A  e.  J  /\  B  e.  J )  ->  ( A  u.  B
)  e.  J )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    u. cun 3411    C_ wss 3413   {cpr 3973   U.cuni 4190   Topctop 19578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rex 2759  df-v 3060  df-un 3418  df-in 3420  df-ss 3427  df-pw 3956  df-sn 3972  df-pr 3974  df-uni 4191  df-top 19583
This theorem is referenced by:  comppfsc  20217  txcld  20288  icccld  21458  icccncfext  37040
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