Theorem ssrabeq 3279

Description: If the restricting class of a restricted class abstraction is a subset of this restricted class abstraction, it is equal to this restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)

Assertion

Ref	Expression
ssrabeq	|- ( V C_ { x e. V | ph } <-> V = { x e. V | ph } )

Distinct variable group:   x, V

Allowed substitution hint:   ph( x)

Proof of Theorem ssrabeq

Step	Hyp	Ref	Expression
1		ssrab2 3277	. . 3 |- { x e. V | ph } C_ V
2	1	biantru 302	. 2 |- ( V C_ { x e. V | ph } <-> ( V C_ { x e. V | ph } /\ { x e. V | ph } C_ V ) )
3		eqss 3207	. 2 |- ( V = { x e. V | ph } <-> ( V C_ { x e. V | ph } /\ { x e. V | ph } C_ V ) )
4	2, 3	bitr4i 187	1 |- ( V C_ { x e. V | ph } <-> V = { x e. V | ph } )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1372   {crab 2487   C_ wss 3165

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rab 2492  df-in 3171  df-ss 3178

This theorem is referenced by:  difrab0eqim  3526