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Theorem uhgr0v0e 40464
 Description: The null graph, with no vertices, has no edges. (Contributed by AV, 21-Oct-2020.)
Hypotheses
Ref Expression
uhgr0v0e.v 𝑉 = (Vtx‘𝐺)
uhgr0v0e.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
uhgr0v0e ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → 𝐸 = ∅)

Proof of Theorem uhgr0v0e
StepHypRef Expression
1 uhgr0v0e.v . . . . . 6 𝑉 = (Vtx‘𝐺)
21eqeq1i 2615 . . . . 5 (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅)
3 uhgr0vb 25738 . . . . . . 7 ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅))
43biimpd 218 . . . . . 6 ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅))
54ex 449 . . . . 5 (𝐺 ∈ UHGraph → ((Vtx‘𝐺) = ∅ → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅)))
62, 5syl5bi 231 . . . 4 (𝐺 ∈ UHGraph → (𝑉 = ∅ → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅)))
76pm2.43a 52 . . 3 (𝐺 ∈ UHGraph → (𝑉 = ∅ → (iEdg‘𝐺) = ∅))
87imp 444 . 2 ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → (iEdg‘𝐺) = ∅)
9 uhgr0v0e.e . . . . 5 𝐸 = (Edg‘𝐺)
109eqeq1i 2615 . . . 4 (𝐸 = ∅ ↔ (Edg‘𝐺) = ∅)
11 uhgriedg0edg0 25801 . . . 4 (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
1210, 11syl5bb 271 . . 3 (𝐺 ∈ UHGraph → (𝐸 = ∅ ↔ (iEdg‘𝐺) = ∅))
1312adantr 480 . 2 ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → (𝐸 = ∅ ↔ (iEdg‘𝐺) = ∅))
148, 13mpbird 246 1 ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → 𝐸 = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∅c0 3874  ‘cfv 5804  Vtxcvtx 25673  iEdgciedg 25674   UHGraph cuhgr 25722  Edgcedga 25792 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-uhgr 25724  df-edga 25793 This theorem is referenced by:  uhgr0vsize0  40465  uhgr0vusgr  40468  fusgrfisbase  40547
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