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Theorem griedg0ssusgr 40489
 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 2680 . . . 4 (𝑔𝑈𝑔 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅})
3 elopab 4908 . . . 4 (𝑔 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ↔ ∃𝑣𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅))
42, 3bitri 263 . . 3 (𝑔𝑈 ↔ ∃𝑣𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅))
5 opex 4859 . . . . . . . 8 𝑣, 𝑒⟩ ∈ V
65a1i 11 . . . . . . 7 (𝑒:∅⟶∅ → ⟨𝑣, 𝑒⟩ ∈ V)
7 vex 3176 . . . . . . . . 9 𝑣 ∈ V
8 vex 3176 . . . . . . . . 9 𝑒 ∈ V
9 opiedgfv 25684 . . . . . . . . 9 ((𝑣 ∈ V ∧ 𝑒 ∈ V) → (iEdg‘⟨𝑣, 𝑒⟩) = 𝑒)
107, 8, 9mp2an 704 . . . . . . . 8 (iEdg‘⟨𝑣, 𝑒⟩) = 𝑒
11 f0bi 6001 . . . . . . . . 9 (𝑒:∅⟶∅ ↔ 𝑒 = ∅)
1211biimpi 205 . . . . . . . 8 (𝑒:∅⟶∅ → 𝑒 = ∅)
1310, 12syl5eq 2656 . . . . . . 7 (𝑒:∅⟶∅ → (iEdg‘⟨𝑣, 𝑒⟩) = ∅)
146, 13usgr0e 40462 . . . . . 6 (𝑒:∅⟶∅ → ⟨𝑣, 𝑒⟩ ∈ USGraph )
1514adantl 481 . . . . 5 ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → ⟨𝑣, 𝑒⟩ ∈ USGraph )
16 eleq1 2676 . . . . . 6 (𝑔 = ⟨𝑣, 𝑒⟩ → (𝑔 ∈ USGraph ↔ ⟨𝑣, 𝑒⟩ ∈ USGraph ))
1716adantr 480 . . . . 5 ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → (𝑔 ∈ USGraph ↔ ⟨𝑣, 𝑒⟩ ∈ USGraph ))
1815, 17mpbird 246 . . . 4 ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → 𝑔 ∈ USGraph )
1918exlimivv 1847 . . 3 (∃𝑣𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → 𝑔 ∈ USGraph )
204, 19sylbi 206 . 2 (𝑔𝑈𝑔 ∈ USGraph )
2120ssriv 3572 1 𝑈 ⊆ USGraph
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ⟨cop 4131  {copab 4642  ⟶wf 5800  ‘cfv 5804  iEdgciedg 25674   USGraph cusgr 40379 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-pow 4769  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-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-f1 5809  df-fv 5812  df-2nd 7060  df-iedg 25676  df-usgr 40381 This theorem is referenced by:  usgrprc  40490
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