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Theorem topopn 11957
Description: The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
Hypothesis
Ref Expression
1open.1  |-  X  = 
U. J
Assertion
Ref Expression
topopn  |-  ( J  e.  Top  ->  X  e.  J )

Proof of Theorem topopn
StepHypRef Expression
1 1open.1 . 2  |-  X  = 
U. J
2 ssid 3067 . . 3  |-  J  C_  J
3 uniopn 11950 . . 3  |-  ( ( J  e.  Top  /\  J  C_  J )  ->  U. J  e.  J
)
42, 3mpan2 419 . 2  |-  ( J  e.  Top  ->  U. J  e.  J )
51, 4syl5eqel 2186 1  |-  ( J  e.  Top  ->  X  e.  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1299    e. wcel 1448    C_ wss 3021   U.cuni 3683   Topctop 11946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-in 3027  df-ss 3034  df-pw 3459  df-uni 3684  df-top 11947
This theorem is referenced by:  toponmax  11974  cldval  12050  ntrfval  12051  clsfval  12052  iscld  12054  ntrval  12061  clsval  12062  0cld  12063  ntrtop  12079  neifval  12091  neif  12092  neival  12094  isnei  12095  tpnei  12111  cnrest  12185  txcn  12225
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