Theorem sselii 3150

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 3149	. 2 |- ( C e. A -> C e. B )
4	1, 3	ax-mp 5	1 |- C e. B

Syntax hints:   e. wcel 2146   C_ wss 3127

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-11 1504  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-in 3133  df-ss 3140

This theorem is referenced by:  brtpos0  6243  ax1cn  7835  recni  7944  0xr  7978  pnfxr  7984  nn0rei  9158  0xnn0  9216  nnzi  9245  nn0zi  9246  lgsdir2lem3  13982