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Theorem eltopss 13171
Description: A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.)
Hypothesis
Ref Expression
1open.1  |-  X  = 
U. J
Assertion
Ref Expression
eltopss  |-  ( ( J  e.  Top  /\  A  e.  J )  ->  A  C_  X )

Proof of Theorem eltopss
StepHypRef Expression
1 elssuni 3835 . . 3  |-  ( A  e.  J  ->  A  C_ 
U. J )
2 1open.1 . . 3  |-  X  = 
U. J
31, 2sseqtrrdi 3204 . 2  |-  ( A  e.  J  ->  A  C_  X )
43adantl 277 1  |-  ( ( J  e.  Top  /\  A  e.  J )  ->  A  C_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148    C_ wss 3129   U.cuni 3807   Topctop 13159
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
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-v 2739  df-in 3135  df-ss 3142  df-uni 3808
This theorem is referenced by:  ntrss3  13287  opnneissb  13319  opnssneib  13320  opnneiss  13322  cnpnei  13383  imasnopn  13463
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