Theorem griedg0prc 26156

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 4790	. . . 4 ⊢ ∅ ∈ V
2		feq1 6026	. . . 4 ⊢ (𝑒 = ∅ → (𝑒:∅⟶∅ ↔ ∅:∅⟶∅))
3		f0 6086	. . . 4 ⊢ ∅:∅⟶∅
4	1, 2, 3	ceqsexv2d 3243	. . 3 ⊢ ∃𝑒 𝑒:∅⟶∅
5		opabn1stprc 7228	. . 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 2902	. . 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 221	1 ⊢ 𝑈 ∉ V

Syntax hints:   ↔ wb 196   = wceq 1483  ∃wex 1704   ∉ wnel 2897  Vcvv 3200  ∅c0 3915  {copab 4712  ⟶wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892

This theorem is referenced by:  usgrprc  26158  rgrusgrprc  26485