Theorem vtxdg0v 16218

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 2298	. . . 4 |- ( U e. V <-> U e. (Vtx ` G ) )
3		fveq2 5648	. . . . . 6 |- ( G = (/) -> (Vtx ` G ) = (Vtx ` (/) ) )
4		vtxval0 15977	. . . . . 6 |- (Vtx ` (/) ) = (/)
5	3, 4	eqtrdi 2280	. . . . 5 |- ( G = (/) -> (Vtx ` G ) = (/) )
6	5	eleq2d 2301	. . . 4 |- ( G = (/) -> ( U e. (Vtx ` G ) <-> U e. (/) ) )
7	2, 6	bitrid 192	. . 3 |- ( G = (/) -> ( U e. V <-> U e. (/) ) )
8		noel 3500	. . . 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 2202   (/)c0 3496   ` cfv 5333   0cc0 8075  Vtxcvtx 15936  VtxDegcvtxdg 16210

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fo 5339  df-fv 5341  df-1st 6312  df-inn 9186  df-ndx 13148  df-slot 13149  df-base 13151  df-vtx 15938

This theorem is referenced by: (None)