Theorem eltopss 12801

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

Step	Hyp	Ref	Expression
1		elssuni 3824	. . 3 |- ( A e. J -> A C_ U. J )
2		1open.1	. . 3 |- X = U. J
3	1, 2	sseqtrrdi 3196	. 2 |- ( A e. J -> A C_ X )
4	3	adantl 275	1 |- ( ( J e. Top /\ A e. J ) -> A C_ X )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   C_ wss 3121   U.cuni 3796   Topctop 12789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-uni 3797

This theorem is referenced by:  ntrss3  12917  opnneissb  12949  opnssneib  12950  opnneiss  12952  cnpnei  13013  imasnopn  13093