Theorem vtxdg0v 16144

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 5639	. . . . . 6 |- ( G = (/) -> (Vtx ` G ) = (Vtx ` (/) ) )
4		vtxval0 15903	. . . . . 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 3498	. . . 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 1397   e. wcel 2202   (/)c0 3494   ` cfv 5326   0cc0 8031  Vtxcvtx 15862  VtxDegcvtxdg 16136

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-1st 6302  df-inn 9143  df-ndx 13084  df-slot 13085  df-base 13087  df-vtx 15864

This theorem is referenced by: (None)