Theorem sselii 3190

Description: Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.)

Hypotheses

Ref	Expression
sseli.1	|- A C_ B
sselii.2	|- C e. A

Assertion

Ref	Expression
sselii	|- C e. B

Proof of Theorem sselii

Step	Hyp	Ref	Expression
1		sselii.2	. 2 |- C e. A
2		sseli.1	. . 3 |- A C_ B
3	2	sseli 3189	. 2 |- ( C e. A -> C e. B )
4	1, 3	ax-mp 5	1 |- C e. B

Syntax hints:   e. wcel 2176   C_ wss 3166

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-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-in 3172  df-ss 3179

This theorem is referenced by:  brtpos0  6338  ax1cn  7974  recni  8084  0xr  8119  pnfxr  8125  nn0rei  9306  0xnn0  9364  nnzi  9393  nn0zi  9394  mincncf  15088  lgsdir2lem3  15507