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Theorem uhgr0v0e 26175
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 2656 . . . . 5 (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅)
3 uhgr0vb 26012 . . . . . . 7 ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅))
43biimpd 219 . . . . . 6 ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅))
54ex 449 . . . . 5 (𝐺 ∈ UHGraph → ((Vtx‘𝐺) = ∅ → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅)))
62, 5syl5bi 232 . . . 4 (𝐺 ∈ UHGraph → (𝑉 = ∅ → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅)))
76pm2.43a 54 . . 3 (𝐺 ∈ UHGraph → (𝑉 = ∅ → (iEdg‘𝐺) = ∅))
87imp 444 . 2 ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → (iEdg‘𝐺) = ∅)
9 uhgr0v0e.e . . . . 5 𝐸 = (Edg‘𝐺)
109eqeq1i 2656 . . . 4 (𝐸 = ∅ ↔ (Edg‘𝐺) = ∅)
11 uhgriedg0edg0 26067 . . . 4 (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
1210, 11syl5bb 272 . . 3 (𝐺 ∈ UHGraph → (𝐸 = ∅ ↔ (iEdg‘𝐺) = ∅))
1312adantr 480 . 2 ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → (𝐸 = ∅ ↔ (iEdg‘𝐺) = ∅))
148, 13mpbird 247 1 ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → 𝐸 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  c0 3948  cfv 5926  Vtxcvtx 25919  iEdgciedg 25920  Edgcedg 25984  UHGraphcuhgr 25996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-edg 25985  df-uhgr 25998
This theorem is referenced by:  uhgr0vsize0  26176  uhgr0vusgr  26179  fusgrfisbase  26265
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