Theorem ssrabeq 3254

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 3252	. . 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 3182	. 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 1363   {crab 2469   C_ wss 3141

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rab 2474  df-in 3147  df-ss 3154

This theorem is referenced by:  difrab0eqim  3501