Theorem ssrab3 3279

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 3278	. 2 |- { x e. A | ph } C_ A
3	1, 2	eqsstri 3225	1 |- B C_ A

Syntax hints:   = wceq 1373   {crab 2488   C_ wss 3166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rab 2493  df-in 3172  df-ss 3179

This theorem is referenced by:  pcprecl  12612  pcprendvds  12613  4sqlem13m  12726  4sqlem14  12727  4sqlem17  12730  nmzsubg  13546  nmznsg  13549  conjnmz  13615  conjnmzb  13616  nzrring  13945  lringnzr  13955  rrgeq0  14027  rrgss  14028  mpodvdsmulf1o  15462  fsumdvdsmul  15463  lgsfcl2  15483