Theorem ssrab3 3287

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 3286	. 2 |- { x e. A | ph } C_ A
3	1, 2	eqsstri 3233	1 |- B C_ A

Syntax hints:   = wceq 1373   {crab 2490   C_ wss 3174

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rab 2495  df-in 3180  df-ss 3187

This theorem is referenced by:  pcprecl  12727  pcprendvds  12728  4sqlem13m  12841  4sqlem14  12842  4sqlem17  12845  nmzsubg  13661  nmznsg  13664  conjnmz  13730  conjnmzb  13731  nzrring  14060  lringnzr  14070  rrgeq0  14142  rrgss  14143  mpodvdsmulf1o  15577  fsumdvdsmul  15578  lgsfcl2  15598