Theorem ssrab3 3334

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 3333	. 2 |- { x e. A | ph } C_ A
3	1, 2	eqsstri 3280	1 |- B C_ A

Syntax hints:   = wceq 1402   {crab 2532   C_ wss 3220

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-in 3226  df-ss 3233

This theorem is referenced by:  if0ss  3639  pcprecl  13046  pcprendvds  13047  4sqlem13m  13160  4sqlem14  13161  4sqlem17  13164  ballotfilemfmpn  13212  ballotfilemafi  13216  ballotfilembfi  13217  ballotfilemth  13259  nmzsubg  13990  nmznsg  13993  conjnmz  14059  conjnmzb  14060  nzrring  14463  lringnzr  14473  rrgeq0  14546  rrgss  14548  psrbagconf1o  14987  mpodvdsmulf1o  16018  fsumdvdsmul  16019  lgsfcl2  16039