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Theorem eltopss 16669
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 3871 . . 3  |-  ( A  e.  J  ->  A  C_ 
U. J )
2 1open.1 . . 3  |-  X  = 
U. J
31, 2syl6sseqr 3238 . 2  |-  ( A  e.  J  ->  A  C_  X )
43adantl 452 1  |-  ( ( J  e.  Top  /\  A  e.  J )  ->  A  C_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165   U.cuni 3843   Topctop 16647
This theorem is referenced by:  riinopn  16670  opncld  16786  ntrval2  16804  ntrss3  16813  cmclsopn  16815  opncldf1  16837  opnneissb  16867  opnssneib  16868  opnneiss  16871  restntr  16928  cnpnei  17009  opnregcld  26351
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-ss 3179  df-uni 3844
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