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Theorem opabn1stprc 8054
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
StepHypRef Expression
1 vex 3473 . . . . . . . 8 𝑥 ∈ V
21biantrur 530 . . . . . . 7 (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
32opabbii 5209 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)}
43dmeqi 5901 . . . . 5 dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)}
5 id 22 . . . . . . 7 (∃𝑦𝜑 → ∃𝑦𝜑)
65ralrimivw 3145 . . . . . 6 (∃𝑦𝜑 → ∀𝑥 ∈ V ∃𝑦𝜑)
7 dmopab3 5916 . . . . . 6 (∀𝑥 ∈ V ∃𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)} = V)
86, 7sylib 217 . . . . 5 (∃𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)} = V)
94, 8eqtrid 2779 . . . 4 (∃𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = V)
10 vprc 5309 . . . . 5 ¬ V ∈ V
1110a1i 11 . . . 4 (∃𝑦𝜑 → ¬ V ∈ V)
129, 11eqneltrd 2848 . . 3 (∃𝑦𝜑 → ¬ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
13 dmexg 7901 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V → dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
1412, 13nsyl 140 . 2 (∃𝑦𝜑 → ¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
15 df-nel 3042 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∉ V ↔ ¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
1614, 15sylibr 233 1 (∃𝑦𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∉ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1534  wex 1774  wcel 2099  wnel 3041  wral 3056  Vcvv 3469  {copab 5204  dom cdm 5672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-nel 3042  df-ral 3057  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-cnv 5680  df-dm 5682  df-rn 5683
This theorem is referenced by:  griedg0prc  29051
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