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

This proof depends on syntax axioms:   = wceq 1402   {crab 2532   C_ wss 3220

This proof depends on 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 proof 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 used by:  if0ss  3642  pcprecl  13068  pcprendvds  13069  4sqlem13m  13182  4sqlem14  13183  4sqlem17  13186  ballotfilemfmpn  13234  ballotfilemafi  13238  ballotfilembfi  13239  ballotfilemth  13281  nmzsubg  14013  nmznsg  14016  conjnmz  14082  conjnmzb  14083  nzrring  14490  lringnzr  14500  rrgeq0  14573  rrgss  14575  psrbagconf1o  15064  mpodvdsmulf1o  16104  fsumdvdsmul  16105  lgsfcl2  16125