Theorem eltopss 7604

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 2530	. . 3 |- (A e. J -> A (_ U.J)
2		1open.1	. . 3 |- X = U.J
3	1, 2	syl6ssr 2111	. 2 |- (A e. J -> A (_ X)
4	3	adantl 390	1 |- ((J e. Top /\ A e. J) -> A (_ X)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960   (_ wss 2050  U.cuni 2507  Topctop 7590

This theorem is referenced by:  opncld 7671  clsval2 7682  ntrval2 7683  ntrss3 7689  cmclsopn 7690  opnneissb 7725  opnssneib 7726  opnneiss 7729  islp2 7744  iscnp2 7758  idcn 7763  homcard 10525

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-in 2054  df-ss 2056  df-uni 2508

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