Theorem griedg0prc 29244

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 5247	. . . 4 ⊢ ∅ ∈ V
2		feq1 6634	. . . 4 ⊢ (𝑒 = ∅ → (𝑒:∅⟶∅ ↔ ∅:∅⟶∅))
3		f0 6709	. . . 4 ⊢ ∅:∅⟶∅
4	1, 2, 3	ceqsexv2d 3488	. . 3 ⊢ ∃𝑒 𝑒:∅⟶∅
5		opabn1stprc 7996	. . 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 3039	. . 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 231	1 ⊢ 𝑈 ∉ V

Syntax hints:   ↔ wb 206   = wceq 1541  ∃wex 1780   ∉ wnel 3033  Vcvv 3437  ∅c0 4282  {copab 5155  ⟶wf 6482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-nel 3034  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-fun 6488  df-fn 6489  df-f 6490

This theorem is referenced by:  usgrprc  29246  rgrusgrprc  29570