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Theorem vtxdg0v 27241
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 2903 . . . 4 (𝑈𝑉𝑈 ∈ (Vtx‘𝐺))
3 fveq2 6643 . . . . . 6 (𝐺 = ∅ → (Vtx‘𝐺) = (Vtx‘∅))
4 vtxval0 26810 . . . . . 6 (Vtx‘∅) = ∅
53, 4syl6eq 2872 . . . . 5 (𝐺 = ∅ → (Vtx‘𝐺) = ∅)
65eleq2d 2897 . . . 4 (𝐺 = ∅ → (𝑈 ∈ (Vtx‘𝐺) ↔ 𝑈 ∈ ∅))
72, 6syl5bb 286 . . 3 (𝐺 = ∅ → (𝑈𝑉𝑈 ∈ ∅))
8 noel 4270 . . . 4 ¬ 𝑈 ∈ ∅
98pm2.21i 119 . . 3 (𝑈 ∈ ∅ → ((VtxDeg‘𝐺)‘𝑈) = 0)
107, 9syl6bi 256 . 2 (𝐺 = ∅ → (𝑈𝑉 → ((VtxDeg‘𝐺)‘𝑈) = 0))
1110imp 410 1 ((𝐺 = ∅ ∧ 𝑈𝑉) → ((VtxDeg‘𝐺)‘𝑈) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  c0 4266  cfv 6328  0cc0 10514  Vtxcvtx 26767  VtxDegcvtxdg 27233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6287  df-fun 6330  df-fv 6336  df-slot 16465  df-base 16467  df-vtx 26769
This theorem is referenced by: (None)
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