Theorem 0vtxrusgr 26473

Description: A graph with no vertices and an empty edge function is a k-regular simple graph for every k. (Contributed by Alexander van der Vekens, 10-Jul-2018.) (Revised by AV, 26-Dec-2020.)

Assertion

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

Distinct variable groups:   k, G   k, W

Proof of Theorem 0vtxrusgr

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgr0v 26133	. . . 4 |- ( ( G e. W /\ (Vtx ` G ) = (/) /\ (iEdg ` G ) = (/) ) -> G e. USGraph )
2	1	adantr 481	. . 3 |- ( ( ( G e. W /\ (Vtx ` G ) = (/) /\ (iEdg ` G ) = (/) ) /\ k e. NN0* ) -> G e. USGraph )
3		0vtxrgr 26472	. . . . . 6 |- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> A. v e. NN0* G RegGraph v )
4		breq2 4657	. . . . . . . 8 |- ( v = k -> ( G RegGraph v <-> G RegGraph k ) )
5	4	rspccva 3308	. . . . . . 7 |- ( ( A. v e. NN0* G RegGraph v /\ k e. NN0* ) -> G RegGraph k )
6	5	ex 450	. . . . . 6 |- ( A. v e. NN0* G RegGraph v -> ( k e. NN0* -> G RegGraph k ) )
7	3, 6	syl 17	. . . . 5 |- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> ( k e. NN0* -> G RegGraph k ) )
8	7	3adant3 1081	. . . 4 |- ( ( G e. W /\ (Vtx ` G ) = (/) /\ (iEdg ` G ) = (/) ) -> ( k e. NN0* -> G RegGraph k ) )
9	8	imp 445	. . 3 |- ( ( ( G e. W /\ (Vtx ` G ) = (/) /\ (iEdg ` G ) = (/) ) /\ k e. NN0* ) -> G RegGraph k )
10		isrusgr 26457	. . . 4 |- ( ( G e. W /\ k e. NN0* ) -> ( G RegUSGraph k <-> ( G e. USGraph /\ G RegGraph k ) ) )
11	10	3ad2antl1 1223	. . 3 |- ( ( ( G e. W /\ (Vtx ` G ) = (/) /\ (iEdg ` G ) = (/) ) /\ k e. NN0* ) -> ( G RegUSGraph k <-> ( G e. USGraph /\ G RegGraph k ) ) )
12	2, 9, 11	mpbir2and 957	. 2 |- ( ( ( G e. W /\ (Vtx ` G ) = (/) /\ (iEdg ` G ) = (/) ) /\ k e. NN0* ) -> G RegUSGraph k )
13	12	ralrimiva 2966	1 |- ( ( G e. W /\ (Vtx ` G ) = (/) /\ (iEdg ` G ) = (/) ) -> A. k e. NN0* G RegUSGraph k )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   (/)c0 3915   class class class wbr 4653   ` cfv 5888  NN0*cxnn0 11363  Vtxcvtx 25874  iEdgciedg 25875   USGraph cusgr 26044   RegGraph crgr 26451   RegUSGraph crusgr 26452

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-2 11079  df-uhgr 25953  df-upgr 25977  df-uspgr 26045  df-usgr 26046  df-rgr 26453  df-rusgr 26454

This theorem is referenced by:  0uhgrrusgr  26474  0grrusgr  26475