Theorem difrab0eqim 3481

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 3234	. 2 |- ( V C_ { x e. V | ph } <-> V = { x e. V | ph } )
2		ssdif0im 3479	. 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 1348   {crab 2452   \ cdif 3118   C_ wss 3121   (/)c0 3414

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415

This theorem is referenced by: (None)