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Theorem usgrprc 40490
 Description: The class of simple graphs is a proper class (and therefore, because of prcssprc 40306, the classes of multigraphs, pseudographs and hypergraphs are proper classes, too). (Contributed by AV, 27-Dec-2020.)
Assertion
Ref Expression
usgrprc USGraph ∉ V

Proof of Theorem usgrprc
Dummy variables 𝑣 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . 3 {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} = {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅}
21griedg0ssusgr 40489 . 2 {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ⊆ USGraph
31griedg0prc 40488 . 2 {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V
4 prcssprc 40306 . 2 (({⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ⊆ USGraph ∧ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V) → USGraph ∉ V)
52, 3, 4mp2an 704 1 USGraph ∉ V
 Colors of variables: wff setvar class Syntax hints:   ∉ wnel 2781  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  {copab 4642  ⟶wf 5800   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-nel 2783  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: (None)
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