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Theorem griedg0prc 40488
 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 4718 . . . 4 ∅ ∈ V
2 feq1 5939 . . . 4 (𝑒 = ∅ → (𝑒:∅⟶∅ ↔ ∅:∅⟶∅))
3 f0 5999 . . . 4 ∅:∅⟶∅
41, 2, 3ceqsexv2d 3216 . . 3 𝑒 𝑒:∅⟶∅
5 opabn1stprc 40318 . . 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 2888 . . 3 (𝑈 = {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} → (𝑈 ∉ V ↔ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V))
97, 8ax-mp 5 . 2 (𝑈 ∉ V ↔ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V)
106, 9mpbir 220 1 𝑈 ∉ V
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475  ∃wex 1695   ∉ wnel 2781  Vcvv 3173  ∅c0 3874  {copab 4642  ⟶wf 5800 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-f 5808 This theorem is referenced by:  usgrprc  40490  rgrusgrprc  40789
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