Theorem ssrab3 3324

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 3323	. 2 |- { x e. A | ph } C_ A
3	1, 2	eqsstri 3270	1 |- B C_ A

Syntax hints:   = wceq 1398   {crab 2524   C_ wss 3211

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rab 2529  df-in 3217  df-ss 3224

This theorem is referenced by:  if0ss  3624  pcprecl  12987  pcprendvds  12988  4sqlem13m  13101  4sqlem14  13102  4sqlem17  13105  nmzsubg  13927  nmznsg  13930  conjnmz  13996  conjnmzb  13997  nzrring  14328  lringnzr  14338  rrgeq0  14410  rrgss  14412  psrbagconf1o  14828  mpodvdsmulf1o  15858  fsumdvdsmul  15859  lgsfcl2  15879