Theorem opabn1stprc 7612

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 3440	. . . . . . . 8 ⊢ 𝑥 ∈ V
2	1	biantrur 531	. . . . . . 7 ⊢ (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
3	2	opabbii 5029	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)}
4	3	dmeqi 5659	. . . . 5 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)}
5		id 22	. . . . . . 7 ⊢ (∃𝑦𝜑 → ∃𝑦𝜑)
6	5	ralrimivw 3150	. . . . . 6 ⊢ (∃𝑦𝜑 → ∀𝑥 ∈ V ∃𝑦𝜑)
7		dmopab3 5674	. . . . . 6 ⊢ (∀𝑥 ∈ V ∃𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)} = V)
8	6, 7	sylib 219	. . . . 5 ⊢ (∃𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)} = V)
9	4, 8	syl5eq 2843	. . . 4 ⊢ (∃𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = V)
10		vprc 5110	. . . . 5 ⊢ ¬ V ∈ V
11	10	a1i 11	. . . 4 ⊢ (∃𝑦𝜑 → ¬ V ∈ V)
12	9, 11	eqneltrd 2902	. . 3 ⊢ (∃𝑦𝜑 → ¬ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
13		dmexg 7469	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V → dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
14	12, 13	nsyl 142	. 2 ⊢ (∃𝑦𝜑 → ¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
15		df-nel 3091	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∉ V ↔ ¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
16	14, 15	sylibr 235	1 ⊢ (∃𝑦𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∉ V)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1522  ∃wex 1761   ∈ wcel 2081   ∉ wnel 3090  ∀wral 3105  Vcvv 3437  {copab 5024  dom cdm 5443

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-nel 3091  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-cnv 5451  df-dm 5453  df-rn 5454

This theorem is referenced by:  griedg0prc  26729