Theorem difrab0eqim 3474

Description: If the difference between the restricting class of a restricted class abstraction and the restricted class abstraction is empty, the restricting class is equal to this restricted class abstraction. (Contributed by Jim Kingdon, 3-Aug-2018.)

Assertion

Ref	Expression
difrab0eqim	|- ( V = { x e. V | ph } -> ( V \ { x e. V | ph } ) = (/) )

Distinct variable group:   x, V

Allowed substitution hint:   ph( x)

Proof of Theorem difrab0eqim

Step	Hyp	Ref	Expression
1		ssrabeq 3228	. 2 |- ( V C_ { x e. V | ph } <-> V = { x e. V | ph } )
2		ssdif0im 3472	. 2 |- ( V C_ { x e. V | ph } -> ( V \ { x e. V | ph } ) = (/) )
3	1, 2	sylbir 134	1 |- ( V = { x e. V | ph } -> ( V \ { x e. V | ph } ) = (/) )

Syntax hints:   -> wi 4   = wceq 1343   {crab 2447   \ cdif 3112   C_ wss 3115   (/)c0 3408

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-in1 604  ax-in2 605  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-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-rab 2452  df-v 2727  df-dif 3117  df-in 3121  df-ss 3128  df-nul 3409

This theorem is referenced by: (None)