Theorem vdif0im 3348

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 3330	. 2 |- ( _V C_ A <-> A = _V )
2		ssdif0im 3347	. 2 |- ( _V C_ A -> ( _V \ A ) = (/) )
3	1, 2	sylbir 133	1 |- ( A = _V -> ( _V \ A ) = (/) )

Syntax hints:   -> wi 4   = wceq 1289   _Vcvv 2619   \ cdif 2996   C_ wss 2999   (/)c0 3286

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 3001  df-in 3005  df-ss 3012  df-nul 3287

This theorem is referenced by: (None)