Theorem opabn1stprc 8055

Description: An ordered-pair class abstraction which does not depend on the first abstraction variable is a proper class. There must be, however, at least one set which satisfies the restricting wff. (Contributed by AV, 27-Dec-2020.)

Assertion

Ref	Expression
opabn1stprc	⊢ (∃𝑦𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∉ V)

Distinct variable groups:   𝑥,𝑦   𝜑,𝑥

Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem opabn1stprc

Step	Hyp	Ref	Expression
1		vex 3467	. . . . . . . 8 ⊢ 𝑥 ∈ V
2	1	biantrur 539	. . . . . . 7 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
3	2	opabbii 5182	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)}
4	3	dmeqi 5895	. . . . 5 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)}
5		id 23	. . . . . . 7 ⊢ (∃𝑦𝜑 → ∃𝑦𝜑)
6	5	ralrimivw 3167	. . . . . 6 ⊢ (∃𝑦𝜑 → ∀𝑥 ∈ V ∃𝑦𝜑)
7		dmopab3 5910	. . . . . 6 ⊢ (∀𝑥 ∈ V ∃𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)} = V)
8	6, 7	sylib 221	. . . . 5 ⊢ (∃𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)} = V)
9	4, 8	eqtrid 2816	. . . 4 ⊢ (∃𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = V)
10		vprc 5285	. . . . 5 ⊢ ¬ V ∈ V
11	10	a1i 11	. . . 4 ⊢ (∃𝑦𝜑 → ¬ V ∈ V)
12	9, 11	eqneltrd 2889	. . 3 ⊢ (∃𝑦𝜑 → ¬ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
13		dmexg 7898	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V → dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
14	12, 13	nsyl 141	. 2 ⊢ (∃𝑦𝜑 → ¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
15		df-nel 3071	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∉ V ↔ ¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
16	14, 15	sylibr 237	1 ⊢ (∃𝑦𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∉ V)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149   ∉ wnel 3070  ∀wral 3085  Vcvv 3463  {copab 5177  dom cdm 5662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-nel 3071  df-ral 3086  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-cnv 5670  df-dm 5672  df-rn 5673

This theorem is referenced by:  griedg0prc  29555