Theorem difrab0eqim 3460

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 3214	. 2 |- ( V C_ { x e. V | ph } <-> V = { x e. V | ph } )
2		ssdif0im 3458	. 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 1335   {crab 2439   \ cdif 3099   C_ wss 3102   (/)c0 3394

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rab 2444  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115  df-nul 3395

This theorem is referenced by: (None)