Theorem ssrab2 3352

Description: Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.)

Assertion

Ref	Expression
ssrab2	⊢ {x ∈ A ∣ φ} ⊆ A

Distinct variable group:   x,A

Allowed substitution hint:   φ(x)

Proof of Theorem ssrab2

Step	Hyp	Ref	Expression
1		df-rab 2624	. 2 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)}
2		ssab2 3351	. 2 ⊢ {x ∣ (x ∈ A ∧ φ)} ⊆ A
3	1, 2	eqsstri 3302	1 ⊢ {x ∈ A ∣ φ} ⊆ A

Syntax hints:   ∧ wa 358   ∈ wcel 1710  {cab 2339  {crab 2619   ⊆ 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-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by:  iinrab2  4030  riinrab  4042  reiotacl  4365  nenpw1pwlem2  6086