Theorem vdif0im 3516

Description: Universal class equality in terms of empty difference. (Contributed by Jim Kingdon, 3-Aug-2018.)

Assertion

Ref	Expression
vdif0im	|- ( A = _V -> ( _V \ A ) = (/) )

Proof of Theorem vdif0im

Step	Hyp	Ref	Expression
1		vss 3498	. 2 |- ( _V C_ A <-> A = _V )
2		ssdif0im 3515	. 2 |- ( _V C_ A -> ( _V \ A ) = (/) )
3	1, 2	sylbir 135	1 |- ( A = _V -> ( _V \ A ) = (/) )

Syntax hints:   -> wi 4   = wceq 1364   _Vcvv 2763   \ cdif 3154   C_ wss 3157   (/)c0 3450

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-in 3163  df-ss 3170  df-nul 3451

This theorem is referenced by: (None)