Theorem ssrabeq 3122

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 3121	. . 3 |- { x e. V | ph } C_ V
2	1	biantru 297	. 2 |- ( V C_ { x e. V | ph } <-> ( V C_ { x e. V | ph } /\ { x e. V | ph } C_ V ) )
3		eqss 3054	. 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 1296   {crab 2374   C_ wss 3013

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rab 2379  df-in 3019  df-ss 3026

This theorem is referenced by:  difrab0eqim  3368