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Theorem topopn 13139
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 3175 . . 3  |-  J  C_  J
3 uniopn 13132 . . 3  |-  ( ( J  e.  Top  /\  J  C_  J )  ->  U. J  e.  J
)
42, 3mpan2 425 . 2  |-  ( J  e.  Top  ->  U. J  e.  J )
51, 4eqeltrid 2264 1  |-  ( J  e.  Top  ->  X  e.  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148    C_ wss 3129   U.cuni 3807   Topctop 13128
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-sep 4118
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-in 3135  df-ss 3142  df-pw 3576  df-uni 3808  df-top 13129
This theorem is referenced by:  toponmax  13156  cldval  13232  ntrfval  13233  clsfval  13234  iscld  13236  ntrval  13243  clsval  13244  0cld  13245  ntrtop  13261  neifval  13273  neif  13274  neival  13276  isnei  13277  tpnei  13293  cnrest  13368  txcn  13408
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