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Theorem griedg0prc 28386
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 5300 . . . 4 ∅ ∈ V
2 feq1 6685 . . . 4 (𝑒 = ∅ → (𝑒:∅⟶∅ ↔ ∅:∅⟶∅))
3 f0 6759 . . . 4 ∅:∅⟶∅
41, 2, 3ceqsexv2d 3525 . . 3 𝑒 𝑒:∅⟶∅
5 opabn1stprc 8026 . . 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 230 1 𝑈 ∉ V
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1541  wex 1781  wnel 3045  Vcvv 3473  c0 4318  {copab 5203  wf 6528
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-fun 6534  df-fn 6535  df-f 6536
This theorem is referenced by:  usgrprc  28388  rgrusgrprc  28711
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