Theorem ssriv 3277

Description: Inference rule based on subclass definition. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
ssriv.1	|- (x e. A -> x e. B)

Assertion

Ref	Expression
ssriv	|- A C_ B

Distinct variable groups:   x,A   x,B

Proof of Theorem ssriv

Step	Hyp	Ref	Expression
1		dfss2 3262	. 2 |- (A C_ B <-> A.x(x e. A -> x e. B))
2		ssriv.1	. 2 |- (x e. A -> x e. B)
3	1, 2	mpgbir 1550	1 |- A C_ 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:  ssid  3290  ssv  3291  difss  3393  ssun1  3426  inss1  3475  0ss  3579  difprsnss  3846  snsspw  3877  uniin  3911  iuniin  3979  iunpwss  4055  cokrelk  4284  evenoddnnnul  4514  dmin  4913  dmcoss  4971  dminss  5041  imainss  5042  nnssnc  6147  cenc  6181