Theorem griedg0prc 29523

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 5262	. . . 4 ⊢ ∅ ∈ V
2		feq1 6673	. . . 4 ⊢ (𝑒 = ∅ → (𝑒:∅⟶∅ ↔ ∅:∅⟶∅))
3		f0 6749	. . . 4 ⊢ ∅:∅⟶∅
4	1, 2, 3	ceqsexv2d 3506	. . 3 ⊢ ∃𝑒 𝑒:∅⟶∅
5		opabn1stprc 8043	. . 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 3070	. . 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

Syntax hints:   ↔ wb 209   = wceq 1563  ∃wex 1802   ∉ wnel 3064  Vcvv 3457  ∅c0 4288  {copab 5167  ⟶wf 6521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-nel 3065  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-fun 6527  df-fn 6528  df-f 6529

This theorem is referenced by:  usgrprc  29525  rgrusgrprc  29848