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Theorem topopn 13993
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 3190 . . 3  |-  J  C_  J
3 uniopn 13986 . . 3  |-  ( ( J  e.  Top  /\  J  C_  J )  ->  U. J  e.  J
)
42, 3mpan2 425 . 2  |-  ( J  e.  Top  ->  U. J  e.  J )
51, 4eqeltrid 2276 1  |-  ( J  e.  Top  ->  X  e.  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 2160    C_ wss 3144   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-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-in 3150  df-ss 3157  df-pw 3595  df-uni 3828  df-top 13983
This theorem is referenced by:  toponmax  14010  cldval  14084  ntrfval  14085  clsfval  14086  iscld  14088  ntrval  14095  clsval  14096  0cld  14097  ntrtop  14113  neifval  14125  neif  14126  neival  14128  isnei  14129  tpnei  14145  cnrest  14220  txcn  14260
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