Theorem ssriv 3278

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

Hypothesis

Ref	Expression
ssriv.1	⊢ (x ∈ A → x ∈ B)

Assertion

Ref	Expression
ssriv	⊢ A ⊆ B

Distinct variable groups:   x,A   x,B

Proof of Theorem ssriv

Step	Hyp	Ref	Expression
1		dfss2 3263	. 2 ⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B))
2		ssriv.1	. 2 ⊢ (x ∈ A → x ∈ B)
3	1, 2	mpgbir 1550	1 ⊢ A ⊆ B

Syntax hints:   → wi 4   ∈ wcel 1710   ⊆ wss 3258

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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by:  ssid  3291  ssv  3292  difss  3394  ssun1  3427  inss1  3476  0ss  3580  difprsnss  3847  snsspw  3878  uniin  3912  iuniin  3980  iunpwss  4056  cokrelk  4285  evenoddnnnul  4515  dmin  4914  dmcoss  4972  dminss  5042  imainss  5043  nnssnc  6148  cenc  6182