Theorem eltopss 15110

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 3963	. . 3 |- ( A e. J -> A C_ U. J )
2		1open.1	. . 3 |- X = U. J
3	1, 2	sseqtrrdi 3297	. 2 |- ( A e. J -> A C_ X )
4	3	adantl 277	1 |- ( ( J e. Top /\ A e. J ) -> A C_ X )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   C_ wss 3220   U.cuni 3935   Topctop 15098

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-uni 3936

This theorem is used by:  ntrss3  15224  opnneissb  15256  opnssneib  15257  opnneiss  15259  cnpnei  15320  imasnopn  15400