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Theorem eltopss 16982
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 4045 . . 3  |-  ( A  e.  J  ->  A  C_ 
U. J )
2 1open.1 . . 3  |-  X  = 
U. J
31, 2syl6sseqr 3397 . 2  |-  ( A  e.  J  ->  A  C_  X )
43adantl 454 1  |-  ( ( J  e.  Top  /\  A  e.  J )  ->  A  C_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    C_ wss 3322   U.cuni 4017   Topctop 16960
This theorem is referenced by:  riinopn  16983  opncld  17099  ntrval2  17117  ntrss3  17126  cmclsopn  17128  opncldf1  17150  opnneissb  17180  opnssneib  17181  opnneiss  17184  neitr  17246  restntr  17248  cnpnei  17330  imasnopn  17724  cnextcn  18100  utopreg  18284  opnregcld  26335
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-in 3329  df-ss 3336  df-uni 4018
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