Theorem ssrab3 3688

Description: Subclass relation for a restricted class abstraction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypothesis

Ref	Expression
ssrab3.1	|- B = { x e. A | ph }

Assertion

Ref	Expression
ssrab3	|- B C_ A

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   B( x)

Proof of Theorem ssrab3

Step	Hyp	Ref	Expression
1		ssrab3.1	. 2 |- B = { x e. A | ph }
2		ssrab2 3687	. 2 |- { x e. A | ph } C_ A
3	1, 2	eqsstri 3635	1 |- B C_ A

Syntax hints:   = wceq 1483   {crab 2916   C_ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-in 3581  df-ss 3588

This theorem is referenced by:  usgrres  26200  frgrwopregbsn  27181  frgrwopreg1  27182  eulerpartlemgvv  30438  reprpmtf1o  30704  hgt750lemb  30734  hgt750leme  30736  bnj1212  30870  bnj213  30952  bnj1286  31087  bnj1312  31126  bnj1523  31139