Theorem eltopss 7603

Description: A member of a topology is a subset of its underlying set.

Hypothesis

Ref	Expression
1open.1	|- X = U.J

Assertion

Ref	Expression
eltopss	|- ((J e. Top /\ A e. J) -> A (_ X)

Proof of Theorem eltopss

Step	Hyp	Ref	Expression
1		elssuni 2526	. . 3 |- (A e. J -> A (_ U.J)
2		1open.1	. . 3 |- X = U.J
3	1, 2	syl6ssr 2108	. 2 |- (A e. J -> A (_ X)
4	3	adantl 388	1 |- ((J e. Top /\ A e. J) -> A (_ X)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958   (_ wss 2047  U.cuni 2503  Topctop 7588

This theorem is referenced by:  opncld 7674  clsval2 7685  ntrval2 7686  ntrss3 7692  cmclsopn 7693  opnneissb 7728  opnssneib 7729  opnneiss 7732  islp2 7747  iscnp2 7761  idcn 7766  homcard 10539

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-in 2051  df-ss 2053  df-uni 2504

Copyright terms: Public domain