Theorem griedg0prc 29732

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

Step	Hyp	Ref	Expression
1		0ex 5268	. . . 4 ⊢ ∅ ∈ V
2		feq1 6684	. . . 4 ⊢ (𝑒 = ∅ → (𝑒:∅⟶∅ ↔ ∅:∅⟶∅))
3		f0 6760	. . . 4 ⊢ ∅:∅⟶∅
4	1, 2, 3	ceqsexv2d 3502	. . 3 ⊢ ∃𝑒 𝑒:∅⟶∅
5		opabn1stprc 8059	. . 3 ⊢ (∃𝑒 𝑒:∅⟶∅ → {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V)
6	4, 5	ax-mp 5	. 2 ⊢ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V
7		griedg0prc.u	. . 3 ⊢ 𝑈 = {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅}
8		neleq1 3069	. . 3 ⊢ (𝑈 = {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} → (𝑈 ∉ V ↔ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V))
9	7, 8	ax-mp 5	. 2 ⊢ (𝑈 ∉ V ↔ {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅} ∉ V)
10	6, 9	mpbir 234	1 ⊢ 𝑈 ∉ V

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  ∃wex 1812   ∉ wnel 3063  Vcvv 3453  ∅c0 4282  {copab 5171  ⟶wf 6533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-nel 3064  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-fun 6539  df-fn 6540  df-f 6541

This theorem is used by:  usgrprc  29734  rgrusgrprc  30057