Theorem difrab0eqim 3434

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 3188	. 2 |- ( V C_ { x e. V | ph } <-> V = { x e. V | ph } )
2		ssdif0im 3432	. 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 1332   {crab 2421   \ cdif 3073   C_ wss 3076   (/)c0 3368

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rab 2426  df-v 2691  df-dif 3078  df-in 3082  df-ss 3089  df-nul 3369

This theorem is referenced by: (None)