Theorem difrab0eqim 3590

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 3336	. 2 |- ( V C_ { x e. V | ph } <-> V = { x e. V | ph } )
2		ssdif0im 3588	. 2 |- ( V C_ { x e. V | ph } -> ( V \ { x e. V | ph } ) = (/) )
3	1, 2	sylbir 135	1 |- ( V = { x e. V | ph } -> ( V \ { x e. V | ph } ) = (/) )

Syntax hints:   -> wi 4   = wceq 1402   {crab 2532   \ cdif 3217   C_ wss 3220   (/)c0 3520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521

This theorem is referenced by: (None)