Theorem sseli 3269

Description: Membership inference from subclass relationship. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
sseli.1	|- A C_ B

Assertion

Ref	Expression
sseli	|- (C e. A -> C e. B)

Proof of Theorem sseli

Step	Hyp	Ref	Expression
1		sseli.1	. 2 |- A C_ B
2		ssel 3267	. 2 |- (A C_ B -> (C e. A -> C e. B))
3	1, 2	ax-mp 8	1 |- (C e. A -> C e. B)

Syntax hints:   -> wi 4   e. wcel 1710   C_ wss 3257

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259

This theorem is referenced by:  sselii  3270  sseldi  3271  elun1  3430  elun2  3431  opkelimagekg  4271  cokrelk  4284  eqpw1uni  4330  dfnnc2  4395  peano5  4409  nnsucelr  4428  vfinspss  4551  phi011lem1  4598  phialllem2  4617  2elresin  5194  fvopab4ndm  5390  fvimacnvi  5402  elpreima  5407  fvmptss  5705  fvmptex  5721  fvmptnf  5723  clos1is  5881  clos1basesuc  5882  erdisj  5972  enadjlem1  6059