Theorem ssrabeq 3026

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 3025	. . 3 |- { x e. V | ph } C_ V
2	1	biantru 286	. 2 |- ( V C_ { x e. V | ph } <-> ( V C_ { x e. V | ph } /\ { x e. V | ph } C_ V ) )
3		eqss 2960	. 2 |- ( V = { x e. V | ph } <-> ( V C_ { x e. V | ph } /\ { x e. V | ph } C_ V ) )
4	2, 3	bitr4i 176	1 |- ( V C_ { x e. V | ph } <-> V = { x e. V | ph } )

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1243   {crab 2310   C_ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-in 2924  df-ss 2931

This theorem is referenced by:  difrab0eqim  3288