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Theorem unopn 13990
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 1017 . 2  |-  ( ( J  e.  Top  /\  A  e.  J  /\  B  e.  J )  ->  U. { A ,  B }  =  ( A  u.  B )
)
3 prssi 3768 . . . 4  |-  ( ( A  e.  J  /\  B  e.  J )  ->  { A ,  B }  C_  J )
4 uniopn 13986 . . . 4  |-  ( ( J  e.  Top  /\  { A ,  B }  C_  J )  ->  U. { A ,  B }  e.  J )
53, 4sylan2 286 . . 3  |-  ( ( J  e.  Top  /\  ( A  e.  J  /\  B  e.  J
) )  ->  U. { A ,  B }  e.  J )
653impb 1201 . 2  |-  ( ( J  e.  Top  /\  A  e.  J  /\  B  e.  J )  ->  U. { A ,  B }  e.  J
)
72, 6eqeltrrd 2267 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 104    /\ w3a 980    = wceq 1364    e. wcel 2160    u. cun 3142    C_ wss 3144   {cpr 3611   U.cuni 3827   Topctop 13982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-sep 4139
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-uni 3828  df-top 13983
This theorem is referenced by:  reopnap  14523
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