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Theorem uhgr0v0e 28228
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 2738 . . . . 5 (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅)
3 uhgr0vb 28065 . . . . . . 7 ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅))
43biimpd 228 . . . . . 6 ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅))
54ex 414 . . . . 5 (𝐺 ∈ UHGraph → ((Vtx‘𝐺) = ∅ → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅)))
62, 5biimtrid 241 . . . 4 (𝐺 ∈ UHGraph → (𝑉 = ∅ → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅)))
76pm2.43a 54 . . 3 (𝐺 ∈ UHGraph → (𝑉 = ∅ → (iEdg‘𝐺) = ∅))
87imp 408 . 2 ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → (iEdg‘𝐺) = ∅)
9 uhgr0v0e.e . . . . 5 𝐸 = (Edg‘𝐺)
109eqeq1i 2738 . . . 4 (𝐸 = ∅ ↔ (Edg‘𝐺) = ∅)
11 uhgriedg0edg0 28120 . . . 4 (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
1210, 11bitrid 283 . . 3 (𝐺 ∈ UHGraph → (𝐸 = ∅ ↔ (iEdg‘𝐺) = ∅))
1312adantr 482 . 2 ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → (𝐸 = ∅ ↔ (iEdg‘𝐺) = ∅))
148, 13mpbird 257 1 ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → 𝐸 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  c0 4283  cfv 6497  Vtxcvtx 27989  iEdgciedg 27990  Edgcedg 28040  UHGraphcuhgr 28049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-edg 28041  df-uhgr 28051
This theorem is referenced by:  uhgr0vsize0  28229  uhgr0vusgr  28232  fusgrfisbase  28318
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