Theorem sseld 3272

Description: Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.)

Hypothesis

Ref	Expression
sseld.1	|- (ph -> A C_ B)

Assertion

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

Proof of Theorem sseld

Step	Hyp	Ref	Expression
1		sseld.1	. 2 |- (ph -> A C_ B)
2		ssel 3267	. 2 |- (A C_ B -> (C e. A -> C e. B))
3	1, 2	syl 15	1 |- (ph -> (C e. A -> C e. B))

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

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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:  sselda  3273  sseldd  3274  ssneld  3275  elelpwi  3732  findsd  4410  sfinltfin  4535  ssbrd  4680  opelf  5235  fun11iun  5305  fvimacnv  5403  ffvelrn  5415  dff3  5420  dff4  5421  enprmaplem3  6078  spacind  6287  dmfrec  6316