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Theorem usgrprc 28256
Description: The class of simple graphs is a proper class (and therefore, because of prcssprc 5283, the classes of multigraphs, pseudographs and hypergraphs are proper classes, too). (Contributed by AV, 27-Dec-2020.)
Assertion
Ref Expression
usgrprc USGraph βˆ‰ V

Proof of Theorem usgrprc
Dummy variables 𝑣 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . 3 {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…} = {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…}
21griedg0ssusgr 28255 . 2 {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…} βŠ† USGraph
31griedg0prc 28254 . 2 {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…} βˆ‰ V
4 prcssprc 5283 . 2 (({βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…} βŠ† USGraph ∧ {βŸ¨π‘£, π‘’βŸ© ∣ 𝑒:βˆ…βŸΆβˆ…} βˆ‰ V) β†’ USGraph βˆ‰ V)
52, 3, 4mp2an 691 1 USGraph βˆ‰ V
Colors of variables: wff setvar class
Syntax hints:   βˆ‰ wnel 3046  Vcvv 3444   βŠ† wss 3911  βˆ…c0 4283  {copab 5168  βŸΆwf 6493  USGraphcusgr 28142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fv 6505  df-2nd 7923  df-iedg 27992  df-usgr 28144
This theorem is referenced by: (None)
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