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Theorem eltopss 16904
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 3986 . . 3  |-  ( A  e.  J  ->  A  C_ 
U. J )
2 1open.1 . . 3  |-  X  = 
U. J
31, 2syl6sseqr 3339 . 2  |-  ( A  e.  J  ->  A  C_  X )
43adantl 453 1  |-  ( ( J  e.  Top  /\  A  e.  J )  ->  A  C_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3264   U.cuni 3958   Topctop 16882
This theorem is referenced by:  riinopn  16905  opncld  17021  ntrval2  17039  ntrss3  17048  cmclsopn  17050  opncldf1  17072  opnneissb  17102  opnssneib  17103  opnneiss  17106  neitr  17167  restntr  17169  cnpnei  17251  imasnopn  17644  cnextcn  18020  utopreg  18204  opnregcld  26025
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-v 2902  df-in 3271  df-ss 3278  df-uni 3959
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