Theorem 0vtxrgr 26472

Description: A null graph (with no vertices) is k-regular for every k. (Contributed by Alexander van der Vekens, 10-Jul-2018.) (Revised by AV, 26-Dec-2020.)

Assertion

Ref	Expression
0vtxrgr	|- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> A. k e. NN0* G RegGraph k )

Distinct variable groups:   k, G   k, W

Proof of Theorem 0vtxrgr

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . 3 |- ( ( ( G e. W /\ (Vtx ` G ) = (/) ) /\ k e. NN0* ) -> k e. NN0* )
2		rzal 4073	. . . 4 |- ( (Vtx ` G ) = (/) -> A. v e. (Vtx ` G ) ( (VtxDeg ` G ) ` v ) = k )
3	2	ad2antlr 763	. . 3 |- ( ( ( G e. W /\ (Vtx ` G ) = (/) ) /\ k e. NN0* ) -> A. v e. (Vtx ` G ) ( (VtxDeg ` G ) ` v ) = k )
4		eqid 2622	. . . . 5 |- (Vtx ` G ) = (Vtx ` G )
5		eqid 2622	. . . . 5 |- (VtxDeg ` G ) = (VtxDeg ` G )
6	4, 5	isrgr 26455	. . . 4 |- ( ( G e. W /\ k e. NN0* ) -> ( G RegGraph k <-> ( k e. NN0* /\ A. v e. (Vtx ` G ) ( (VtxDeg ` G ) ` v ) = k ) ) )
7	6	adantlr 751	. . 3 |- ( ( ( G e. W /\ (Vtx ` G ) = (/) ) /\ k e. NN0* ) -> ( G RegGraph k <-> ( k e. NN0* /\ A. v e. (Vtx ` G ) ( (VtxDeg ` G ) ` v ) = k ) ) )
8	1, 3, 7	mpbir2and 957	. 2 |- ( ( ( G e. W /\ (Vtx ` G ) = (/) ) /\ k e. NN0* ) -> G RegGraph k )
9	8	ralrimiva 2966	1 |- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> A. k e. NN0* G RegGraph k )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   (/)c0 3915   class class class wbr 4653   ` cfv 5888  NN0*cxnn0 11363  Vtxcvtx 25874  VtxDegcvtxdg 26361   RegGraph crgr 26451

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ne 2795  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-opab 4713  df-iota 5851  df-fv 5896  df-rgr 26453

This theorem is referenced by:  0vtxrusgr  26473