Theorem sselii 3198

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 3197	. 2 |- ( C e. A -> C e. B )
4	1, 3	ax-mp 5	1 |- C e. B

Syntax hints:   e. wcel 2178   C_ wss 3174

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-in 3180  df-ss 3187

This theorem is referenced by:  brtpos0  6361  ax1cn  8009  recni  8119  0xr  8154  pnfxr  8160  nn0rei  9341  0xnn0  9399  nnzi  9428  nn0zi  9429  mincncf  15203  lgsdir2lem3  15622