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Theorem opabn1stprc 7213
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 wwf. (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 3198 . . . . . . . 8 𝑥 ∈ V
21biantrur 527 . . . . . . 7 (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
32opabbii 4708 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)}
43dmeqi 5314 . . . . 5 dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)}
5 id 22 . . . . . . 7 (∃𝑦𝜑 → ∃𝑦𝜑)
65ralrimivw 2964 . . . . . 6 (∃𝑦𝜑 → ∀𝑥 ∈ V ∃𝑦𝜑)
7 dmopab3 5326 . . . . . 6 (∀𝑥 ∈ V ∃𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)} = V)
86, 7sylib 208 . . . . 5 (∃𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)} = V)
94, 8syl5eq 2666 . . . 4 (∃𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = V)
10 vprc 4787 . . . . 5 ¬ V ∈ V
1110a1i 11 . . . 4 (∃𝑦𝜑 → ¬ V ∈ V)
129, 11eqneltrd 2718 . . 3 (∃𝑦𝜑 → ¬ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
13 dmexg 7082 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V → dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
1412, 13nsyl 135 . 2 (∃𝑦𝜑 → ¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
15 df-nel 2895 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∉ V ↔ ¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
1614, 15sylibr 224 1 (∃𝑦𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∉ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1481  wex 1702  wcel 1988  wnel 2894  wral 2909  Vcvv 3195  {copab 4703  dom cdm 5104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-nel 2895  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-cnv 5112  df-dm 5114  df-rn 5115
This theorem is referenced by:  griedg0prc  26137
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