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Theorem griedg0prc 27352
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 5200 . . . 4 ∅ ∈ V
2 feq1 6526 . . . 4 (𝑒 = ∅ → (𝑒:∅⟶∅ ↔ ∅:∅⟶∅))
3 f0 6600 . . . 4 ∅:∅⟶∅
41, 2, 3ceqsexv2d 3457 . . 3 𝑒 𝑒:∅⟶∅
5 opabn1stprc 7828 . . 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 3051 . . 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 1543  wex 1787  wnel 3046  Vcvv 3408  c0 4237  {copab 5115  wf 6376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-fun 6382  df-fn 6383  df-f 6384
This theorem is referenced by:  usgrprc  27354  rgrusgrprc  27677
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