Theorem ssrab3 3311

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 3310	. 2 |- { x e. A | ph } C_ A
3	1, 2	eqsstri 3257	1 |- B C_ A

Syntax hints:   = wceq 1395   {crab 2512   C_ wss 3198

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-in 3204  df-ss 3211

This theorem is referenced by:  pcprecl  12852  pcprendvds  12853  4sqlem13m  12966  4sqlem14  12967  4sqlem17  12970  nmzsubg  13787  nmznsg  13790  conjnmz  13856  conjnmzb  13857  nzrring  14187  lringnzr  14197  rrgeq0  14269  rrgss  14270  mpodvdsmulf1o  15704  fsumdvdsmul  15705  lgsfcl2  15725