Theorem ssrab3 3265

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 3264	. 2 |- { x e. A | ph } C_ A
3	1, 2	eqsstri 3211	1 |- B C_ A

Syntax hints:   = wceq 1364   {crab 2476   C_ wss 3153

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-in 3159  df-ss 3166

This theorem is referenced by:  pcprecl  12427  pcprendvds  12428  4sqlem13m  12541  4sqlem14  12542  4sqlem17  12545  nmzsubg  13280  nmznsg  13283  conjnmz  13349  conjnmzb  13350  nzrring  13679  lringnzr  13689  rrgeq0  13761  rrgss  13762  lgsfcl2  15122