Theorem opabn1stprc 7750

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 3497	. . . . . . . 8 ⊢ 𝑥 ∈ V
2	1	biantrur 533	. . . . . . 7 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
3	2	opabbii 5125	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)}
4	3	dmeqi 5767	. . . . 5 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)}
5		id 22	. . . . . . 7 ⊢ (∃𝑦𝜑 → ∃𝑦𝜑)
6	5	ralrimivw 3183	. . . . . 6 ⊢ (∃𝑦𝜑 → ∀𝑥 ∈ V ∃𝑦𝜑)
7		dmopab3 5782	. . . . . 6 ⊢ (∀𝑥 ∈ V ∃𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)} = V)
8	6, 7	sylib 220	. . . . 5 ⊢ (∃𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)} = V)
9	4, 8	syl5eq 2868	. . . 4 ⊢ (∃𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = V)
10		vprc 5211	. . . . 5 ⊢ ¬ V ∈ V
11	10	a1i 11	. . . 4 ⊢ (∃𝑦𝜑 → ¬ V ∈ V)
12	9, 11	eqneltrd 2932	. . 3 ⊢ (∃𝑦𝜑 → ¬ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
13		dmexg 7607	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V → dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
14	12, 13	nsyl 142	. 2 ⊢ (∃𝑦𝜑 → ¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
15		df-nel 3124	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∉ V ↔ ¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
16	14, 15	sylibr 236	1 ⊢ (∃𝑦𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∉ V)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110   ∉ wnel 3123  ∀wral 3138  Vcvv 3494  {copab 5120  dom cdm 5549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-nel 3124  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-cnv 5557  df-dm 5559  df-rn 5560

This theorem is referenced by:  griedg0prc  27040