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Theorem vtxdg0v 26604
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
vtxdgf.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
vtxdg0v ((𝐺 = ∅ ∧ 𝑈𝑉) → ((VtxDeg‘𝐺)‘𝑈) = 0)

Proof of Theorem vtxdg0v
StepHypRef Expression
1 vtxdgf.v . . . . 5 𝑉 = (Vtx‘𝐺)
21eleq2i 2842 . . . 4 (𝑈𝑉𝑈 ∈ (Vtx‘𝐺))
3 fveq2 6333 . . . . . 6 (𝐺 = ∅ → (Vtx‘𝐺) = (Vtx‘∅))
4 vtxval0 26152 . . . . . 6 (Vtx‘∅) = ∅
53, 4syl6eq 2821 . . . . 5 (𝐺 = ∅ → (Vtx‘𝐺) = ∅)
65eleq2d 2836 . . . 4 (𝐺 = ∅ → (𝑈 ∈ (Vtx‘𝐺) ↔ 𝑈 ∈ ∅))
72, 6syl5bb 272 . . 3 (𝐺 = ∅ → (𝑈𝑉𝑈 ∈ ∅))
8 noel 4067 . . . 4 ¬ 𝑈 ∈ ∅
98pm2.21i 117 . . 3 (𝑈 ∈ ∅ → ((VtxDeg‘𝐺)‘𝑈) = 0)
107, 9syl6bi 243 . 2 (𝐺 = ∅ → (𝑈𝑉 → ((VtxDeg‘𝐺)‘𝑈) = 0))
1110imp 393 1 ((𝐺 = ∅ ∧ 𝑈𝑉) → ((VtxDeg‘𝐺)‘𝑈) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  c0 4063  cfv 6030  0cc0 10142  Vtxcvtx 26095  VtxDegcvtxdg 26596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-slot 16068  df-base 16070  df-vtx 26097
This theorem is referenced by: (None)
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