Theorem sselii 2066

Description: Membership inference from subclass relationship.

Hypotheses

Ref	Expression
sseli.1	|- A (_ 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 (_ B
3	2	sseli 2065	. 2 |- (C e. A -> C e. B)
4	1, 3	ax-mp 7	1 |- C e. B

Syntax hints:   e. wcel 958   (_ wss 2047

This theorem is referenced by:  tz7.44-1 3928  tz7.44-2 3929  alephfp2 4901  ax1cn 5269  recn 5314  nn0re 6110  cvg3 6923  clm4 7080  ivthlem9 7289  isupivth 7290  dsupivthlem 7291  minvecle 8586  sheli 9083  cheli 9103  omlsilem 9244  nonbool 9596  pjssm 9626  riesz4t 9997  riesz1t 9998  cnlnadjeut 10011  nmopadjle 10021  adjeq0 10024

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-in 2051  df-ss 2053

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