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Theorem uhgr0v0e 27031
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 2806 . . . . 5 (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅)
3 uhgr0vb 26868 . . . . . . 7 ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅))
43biimpd 232 . . . . . 6 ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅))
54ex 416 . . . . 5 (𝐺 ∈ UHGraph → ((Vtx‘𝐺) = ∅ → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅)))
62, 5syl5bi 245 . . . 4 (𝐺 ∈ UHGraph → (𝑉 = ∅ → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅)))
76pm2.43a 54 . . 3 (𝐺 ∈ UHGraph → (𝑉 = ∅ → (iEdg‘𝐺) = ∅))
87imp 410 . 2 ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → (iEdg‘𝐺) = ∅)
9 uhgr0v0e.e . . . . 5 𝐸 = (Edg‘𝐺)
109eqeq1i 2806 . . . 4 (𝐸 = ∅ ↔ (Edg‘𝐺) = ∅)
11 uhgriedg0edg0 26923 . . . 4 (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
1210, 11syl5bb 286 . . 3 (𝐺 ∈ UHGraph → (𝐸 = ∅ ↔ (iEdg‘𝐺) = ∅))
1312adantr 484 . 2 ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → (𝐸 = ∅ ↔ (iEdg‘𝐺) = ∅))
148, 13mpbird 260 1 ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → 𝐸 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  c0 4246  cfv 6328  Vtxcvtx 26792  iEdgciedg 26793  Edgcedg 26843  UHGraphcuhgr 26852
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-edg 26844  df-uhgr 26854
This theorem is referenced by:  uhgr0vsize0  27032  uhgr0vusgr  27035  fusgrfisbase  27121
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