Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  usgrprc Structured version   Visualization version   GIF version

Theorem usgrprc 27155
 Description: The class of simple graphs is a proper class (and therefore, because of prcssprc 5195, 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 2758 . . 3 {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} = {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅}
21griedg0ssusgr 27154 . 2 {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ⊆ USGraph
31griedg0prc 27153 . 2 {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V
4 prcssprc 5195 . 2 (({⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ⊆ USGraph ∧ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V) → USGraph ∉ V)
52, 3, 4mp2an 691 1 USGraph ∉ V
 Colors of variables: wff setvar class Syntax hints:   ∉ wnel 3055  Vcvv 3409   ⊆ wss 3858  ∅c0 4225  {copab 5094  ⟶wf 6331  USGraphcusgr 27041 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 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fv 6343  df-2nd 7694  df-iedg 26891  df-usgr 27043 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator