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Theorem griedg0ssusgr 27099
 Description: The class of all simple graphs is a superclass of the class of empty graphs represented as ordered pairs. (Contributed by AV, 27-Dec-2020.)
Hypothesis
Ref Expression
griedg0prc.u 𝑈 = {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅}
Assertion
Ref Expression
griedg0ssusgr 𝑈 ⊆ USGraph
Distinct variable group:   𝑣,𝑒
Allowed substitution hints:   𝑈(𝑣,𝑒)

Proof of Theorem griedg0ssusgr
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 griedg0prc.u . . . . 5 𝑈 = {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅}
21eleq2i 2881 . . . 4 (𝑔𝑈𝑔 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅})
3 elopab 5383 . . . 4 (𝑔 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ↔ ∃𝑣𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅))
42, 3bitri 278 . . 3 (𝑔𝑈 ↔ ∃𝑣𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅))
5 opex 5325 . . . . . . . 8 𝑣, 𝑒⟩ ∈ V
65a1i 11 . . . . . . 7 (𝑒:∅⟶∅ → ⟨𝑣, 𝑒⟩ ∈ V)
7 vex 3445 . . . . . . . . 9 𝑣 ∈ V
8 vex 3445 . . . . . . . . 9 𝑒 ∈ V
97, 8opiedgfvi 26847 . . . . . . . 8 (iEdg‘⟨𝑣, 𝑒⟩) = 𝑒
10 f0bi 6544 . . . . . . . . 9 (𝑒:∅⟶∅ ↔ 𝑒 = ∅)
1110biimpi 219 . . . . . . . 8 (𝑒:∅⟶∅ → 𝑒 = ∅)
129, 11syl5eq 2845 . . . . . . 7 (𝑒:∅⟶∅ → (iEdg‘⟨𝑣, 𝑒⟩) = ∅)
136, 12usgr0e 27070 . . . . . 6 (𝑒:∅⟶∅ → ⟨𝑣, 𝑒⟩ ∈ USGraph)
1413adantl 485 . . . . 5 ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → ⟨𝑣, 𝑒⟩ ∈ USGraph)
15 eleq1 2877 . . . . . 6 (𝑔 = ⟨𝑣, 𝑒⟩ → (𝑔 ∈ USGraph ↔ ⟨𝑣, 𝑒⟩ ∈ USGraph))
1615adantr 484 . . . . 5 ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → (𝑔 ∈ USGraph ↔ ⟨𝑣, 𝑒⟩ ∈ USGraph))
1714, 16mpbird 260 . . . 4 ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → 𝑔 ∈ USGraph)
1817exlimivv 1933 . . 3 (∃𝑣𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → 𝑔 ∈ USGraph)
194, 18sylbi 220 . 2 (𝑔𝑈𝑔 ∈ USGraph)
2019ssriv 3921 1 𝑈 ⊆ USGraph
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3442   ⊆ wss 3883  ∅c0 4246  ⟨cop 4534  {copab 5096  ⟶wf 6328  ‘cfv 6332  iEdgciedg 26834  USGraphcusgr 26986 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299  ax-un 7454 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fv 6340  df-2nd 7685  df-iedg 26836  df-usgr 26988 This theorem is referenced by:  usgrprc  27100
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