Theorem eltopss 14677

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 3915	. . 3 |- ( A e. J -> A C_ U. J )
2		1open.1	. . 3 |- X = U. J
3	1, 2	sseqtrrdi 3273	. 2 |- ( A e. J -> A C_ X )
4	3	adantl 277	1 |- ( ( J e. Top /\ A e. J ) -> A C_ X )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   C_ wss 3197   U.cuni 3887   Topctop 14665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210  df-uni 3888

This theorem is referenced by:  ntrss3  14791  opnneissb  14823  opnssneib  14824  opnneiss  14826  cnpnei  14887  imasnopn  14967