Theorem vtxvalprc 25937

Description: Degenerated case 4 for vertices: The set of vertices of a proper class is the empty set. (Contributed by AV, 12-Oct-2020.)

Assertion

Ref	Expression
vtxvalprc	|- ( C e/ _V -> (Vtx ` C ) = (/) )

Proof of Theorem vtxvalprc

Step	Hyp	Ref	Expression
1		df-nel 2898	. 2 |- ( C e/ _V <-> -. C e. _V )
2		fvprc 6185	. 2 |- ( -. C e. _V -> (Vtx ` C ) = (/) )
3	1, 2	sylbi 207	1 |- ( C e/ _V -> (Vtx ` C ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   e/ wnel 2897   _Vcvv 3200   (/)c0 3915   ` cfv 5888  Vtxcvtx 25874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789  ax-pow 4843

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  wlk0prc  26550