Theorem vtxdg0v 16449

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 2305	. . . 4 |- ( U e. V <-> U e. (Vtx ` G ) )
3		fveq2 5690	. . . . . 6 |- ( G = (/) -> (Vtx ` G ) = (Vtx ` (/) ) )
4		vtxval0 16208	. . . . . 6 |- (Vtx ` (/) ) = (/)
5	3, 4	eqtrdi 2287	. . . . 5 |- ( G = (/) -> (Vtx ` G ) = (/) )
6	5	eleq2d 2308	. . . 4 |- ( G = (/) -> ( U e. (Vtx ` G ) <-> U e. (/) ) )
7	2, 6	bitrid 192	. . 3 |- ( G = (/) -> ( U e. V <-> U e. (/) ) )
8		noel 3525	. . . 4 |- -. U e. (/)
9	8	pm2.21i 655	. . 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 1402   e. wcel 2209   (/)c0 3520   ` cfv 5372   0cc0 8169  Vtxcvtx 16167  VtxDegcvtxdg 16441

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fo 5378  df-fv 5380  df-1st 6364  df-inn 9284  df-ndx 13333  df-slot 13334  df-base 13336  df-vtx 16169

This theorem is referenced by: (None)