Theorem sselii 3153

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 3152	. 2 |- ( C e. A -> C e. B )
4	1, 3	ax-mp 5	1 |- C e. B

Syntax hints:   e. wcel 2148   C_ wss 3130

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3136  df-ss 3143

This theorem is referenced by:  brtpos0  6253  ax1cn  7860  recni  7969  0xr  8004  pnfxr  8010  nn0rei  9187  0xnn0  9245  nnzi  9274  nn0zi  9275  lgsdir2lem3  14434