Theorem opabn1stprc 8011

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 3433	. . . . . . . 8 ⊢ 𝑥 ∈ V
2	1	biantrur 530	. . . . . . 7 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
3	2	opabbii 5152	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)}
4	3	dmeqi 5859	. . . . 5 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)}
5		id 22	. . . . . . 7 ⊢ (∃𝑦𝜑 → ∃𝑦𝜑)
6	5	ralrimivw 3133	. . . . . 6 ⊢ (∃𝑦𝜑 → ∀𝑥 ∈ V ∃𝑦𝜑)
7		dmopab3 5874	. . . . . 6 ⊢ (∀𝑥 ∈ V ∃𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)} = V)
8	6, 7	sylib 218	. . . . 5 ⊢ (∃𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)} = V)
9	4, 8	eqtrid 2783	. . . 4 ⊢ (∃𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = V)
10		vprc 5255	. . . . 5 ⊢ ¬ V ∈ V
11	10	a1i 11	. . . 4 ⊢ (∃𝑦𝜑 → ¬ V ∈ V)
12	9, 11	eqneltrd 2856	. . 3 ⊢ (∃𝑦𝜑 → ¬ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
13		dmexg 7852	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V → dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
14	12, 13	nsyl 140	. 2 ⊢ (∃𝑦𝜑 → ¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
15		df-nel 3037	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∉ V ↔ ¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
16	14, 15	sylibr 234	1 ⊢ (∃𝑦𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∉ V)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ∉ wnel 3036  ∀wral 3051  Vcvv 3429  {copab 5147  dom cdm 5631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-nel 3037  df-ral 3052  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-cnv 5639  df-dm 5641  df-rn 5642

This theorem is referenced by:  griedg0prc  29333