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Theorem opabn1stprc 40238
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 3080 . . . . . . . 8 𝑥 ∈ V
21biantrur 525 . . . . . . 7 (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
32opabbii 4547 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)}
43dmeqi 5138 . . . . 5 dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)}
5 id 22 . . . . . . 7 (∃𝑦𝜑 → ∃𝑦𝜑)
65ralrimivw 2854 . . . . . 6 (∃𝑦𝜑 → ∀𝑥 ∈ V ∃𝑦𝜑)
7 dmopab3 5150 . . . . . 6 (∀𝑥 ∈ V ∃𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)} = V)
86, 7sylib 206 . . . . 5 (∃𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)} = V)
94, 8syl5eq 2560 . . . 4 (∃𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = V)
10 vprc 4623 . . . . 5 ¬ V ∈ V
1110a1i 11 . . . 4 (∃𝑦𝜑 → ¬ V ∈ V)
129, 11eqneltrd 2611 . . 3 (∃𝑦𝜑 → ¬ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
13 dmexg 6863 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V → dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
1412, 13nsyl 133 . 2 (∃𝑦𝜑 → ¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
15 df-nel 2687 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∉ V ↔ ¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
1614, 15sylibr 222 1 (∃𝑦𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∉ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wex 1694  wcel 1938  wnel 2685  wral 2800  Vcvv 3077  {copab 4540  dom cdm 4932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732  ax-un 6721
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-nel 2687  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-cnv 4940  df-dm 4942  df-rn 4943
This theorem is referenced by:  griedg0prc  40595
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