Theorem sselii 3245

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 3244	. 2 |- ( C e. A -> C e. B )
4	1, 3	ax-mp 5	1 |- C e. B

Syntax hints:   e. wcel 2209   C_ wss 3220

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233

This theorem is referenced by:  brtpos0  6513  ax1cn  8218  recni  8328  0xr  8362  pnfxr  8368  nn0rei  9553  0xnn0  9615  nnzi  9644  nn0zi  9645  hashfibclem  11260  gsumclfi  14136  gsummptfidmadd  14138  gsumsubmclfi  14140  gsumfsum  14895  mincncf  15640  lgsdir2lem3  16063