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Theorem griedg0prc 27095
 Description: The class of empty graphs (represented as ordered pairs) is a proper class. (Contributed by AV, 27-Dec-2020.)
Hypothesis
Ref Expression
griedg0prc.u 𝑈 = {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅}
Assertion
Ref Expression
griedg0prc 𝑈 ∉ V
Distinct variable group:   𝑣,𝑒
Allowed substitution hints:   𝑈(𝑣,𝑒)

Proof of Theorem griedg0prc
StepHypRef Expression
1 0ex 5178 . . . 4 ∅ ∈ V
2 feq1 6473 . . . 4 (𝑒 = ∅ → (𝑒:∅⟶∅ ↔ ∅:∅⟶∅))
3 f0 6539 . . . 4 ∅:∅⟶∅
41, 2, 3ceqsexv2d 3490 . . 3 𝑒 𝑒:∅⟶∅
5 opabn1stprc 7748 . . 3 (∃𝑒 𝑒:∅⟶∅ → {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V)
64, 5ax-mp 5 . 2 {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V
7 griedg0prc.u . . 3 𝑈 = {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅}
8 neleq1 3096 . . 3 (𝑈 = {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} → (𝑈 ∉ V ↔ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V))
97, 8ax-mp 5 . 2 (𝑈 ∉ V ↔ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V)
106, 9mpbir 234 1 𝑈 ∉ V
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538  ∃wex 1781   ∉ wnel 3091  Vcvv 3441  ∅c0 4245  {copab 5095  ⟶wf 6325 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 2770  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7451 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-nel 3092  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4246  df-if 4428  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-fun 6331  df-fn 6332  df-f 6333 This theorem is referenced by:  usgrprc  27097  rgrusgrprc  27420
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