Theorem vtxdg0v 16415

Description: The degree of a vertex in the null graph is zero (or anything else), because there are no vertices. (Contributed by AV, 11-Dec-2020.)

Hypothesis

Ref	Expression
vtxdg0v.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
vtxdg0v	|- ( ( G = (/) /\ U e. V ) -> ( (VtxDeg ` G ) ` U ) = 0 )

Proof of Theorem vtxdg0v

Step	Hyp	Ref	Expression
1		vtxdg0v.v	. . . . 5 |- V = (Vtx ` G )
2	1	eleq2i 2301	. . . 4 |- ( U e. V <-> U e. (Vtx ` G ) )
3		fveq2 5675	. . . . . 6 |- ( G = (/) -> (Vtx ` G ) = (Vtx ` (/) ) )
4		vtxval0 16174	. . . . . 6 |- (Vtx ` (/) ) = (/)
5	3, 4	eqtrdi 2283	. . . . 5 |- ( G = (/) -> (Vtx ` G ) = (/) )
6	5	eleq2d 2304	. . . 4 |- ( G = (/) -> ( U e. (Vtx ` G ) <-> U e. (/) ) )
7	2, 6	bitrid 192	. . 3 |- ( G = (/) -> ( U e. V <-> U e. (/) ) )
8		noel 3516	. . . 4 |- -. U e. (/)
9	8	pm2.21i 651	. . 3 |- ( U e. (/) -> ( (VtxDeg ` G ) ` U ) = 0 )
10	7, 9	biimtrdi 163	. 2 |- ( G = (/) -> ( U e. V -> ( (VtxDeg ` G ) ` U ) = 0 ) )
11	10	imp 124	1 |- ( ( G = (/) /\ U e. V ) -> ( (VtxDeg ` G ) ` U ) = 0 )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   (/)c0 3512   ` cfv 5357   0cc0 8143  Vtxcvtx 16133  VtxDegcvtxdg 16407

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fo 5363  df-fv 5365  df-1st 6347  df-inn 9255  df-ndx 13299  df-slot 13300  df-base 13302  df-vtx 16135

This theorem is referenced by: (None)