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Theorem griedg0ssusgr 26050
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 2690 . . . 4 (𝑔𝑈𝑔 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅})
3 elopab 4943 . . . 4 (𝑔 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ↔ ∃𝑣𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅))
42, 3bitri 264 . . 3 (𝑔𝑈 ↔ ∃𝑣𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅))
5 opex 4893 . . . . . . . 8 𝑣, 𝑒⟩ ∈ V
65a1i 11 . . . . . . 7 (𝑒:∅⟶∅ → ⟨𝑣, 𝑒⟩ ∈ V)
7 vex 3189 . . . . . . . . 9 𝑣 ∈ V
8 vex 3189 . . . . . . . . 9 𝑒 ∈ V
9 opiedgfv 25787 . . . . . . . . 9 ((𝑣 ∈ V ∧ 𝑒 ∈ V) → (iEdg‘⟨𝑣, 𝑒⟩) = 𝑒)
107, 8, 9mp2an 707 . . . . . . . 8 (iEdg‘⟨𝑣, 𝑒⟩) = 𝑒
11 f0bi 6045 . . . . . . . . 9 (𝑒:∅⟶∅ ↔ 𝑒 = ∅)
1211biimpi 206 . . . . . . . 8 (𝑒:∅⟶∅ → 𝑒 = ∅)
1310, 12syl5eq 2667 . . . . . . 7 (𝑒:∅⟶∅ → (iEdg‘⟨𝑣, 𝑒⟩) = ∅)
146, 13usgr0e 26021 . . . . . 6 (𝑒:∅⟶∅ → ⟨𝑣, 𝑒⟩ ∈ USGraph )
1514adantl 482 . . . . 5 ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → ⟨𝑣, 𝑒⟩ ∈ USGraph )
16 eleq1 2686 . . . . . 6 (𝑔 = ⟨𝑣, 𝑒⟩ → (𝑔 ∈ USGraph ↔ ⟨𝑣, 𝑒⟩ ∈ USGraph ))
1716adantr 481 . . . . 5 ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → (𝑔 ∈ USGraph ↔ ⟨𝑣, 𝑒⟩ ∈ USGraph ))
1815, 17mpbird 247 . . . 4 ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → 𝑔 ∈ USGraph )
1918exlimivv 1857 . . 3 (∃𝑣𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:∅⟶∅) → 𝑔 ∈ USGraph )
204, 19sylbi 207 . 2 (𝑔𝑈𝑔 ∈ USGraph )
2120ssriv 3587 1 𝑈 ⊆ USGraph
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  Vcvv 3186  wss 3555  c0 3891  cop 4154  {copab 4672  wf 5843  cfv 5847  iEdgciedg 25775   USGraph cusgr 25937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fv 5855  df-2nd 7114  df-iedg 25777  df-usgr 25939
This theorem is referenced by:  usgrprc  26051
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