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Theorem usgrprc 28956
Description: The class of simple graphs is a proper class (and therefore, because of prcssprc 5325, 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 2731 . . 3 {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} = {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅}
21griedg0ssusgr 28955 . 2 {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ⊆ USGraph
31griedg0prc 28954 . 2 {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V
4 prcssprc 5325 . 2 (({⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ⊆ USGraph ∧ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V) → USGraph ∉ V)
52, 3, 4mp2an 689 1 USGraph ∉ V
Colors of variables: wff setvar class
Syntax hints:  wnel 3045  Vcvv 3473  wss 3948  c0 4322  {copab 5210  wf 6539  USGraphcusgr 28842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fv 6551  df-2nd 7980  df-iedg 28692  df-usgr 28844
This theorem is referenced by: (None)
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