Theorem ssrabeq 3228

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 3226	. . 3 |- { x e. V | ph } C_ V
2	1	biantru 300	. 2 |- ( V C_ { x e. V | ph } <-> ( V C_ { x e. V | ph } /\ { x e. V | ph } C_ V ) )
3		eqss 3156	. 2 |- ( V = { x e. V | ph } <-> ( V C_ { x e. V | ph } /\ { x e. V | ph } C_ V ) )
4	2, 3	bitr4i 186	1 |- ( V C_ { x e. V | ph } <-> V = { x e. V | ph } )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1343   {crab 2447   C_ wss 3115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-rab 2452  df-in 3121  df-ss 3128

This theorem is referenced by:  difrab0eqim  3474