MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opabn1stprc Structured version   Visualization version   GIF version

Theorem opabn1stprc 7742
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 3447 . . . . . . . 8 𝑥 ∈ V
21biantrur 534 . . . . . . 7 (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
32opabbii 5100 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)}
43dmeqi 5741 . . . . 5 dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)}
5 id 22 . . . . . . 7 (∃𝑦𝜑 → ∃𝑦𝜑)
65ralrimivw 3153 . . . . . 6 (∃𝑦𝜑 → ∀𝑥 ∈ V ∃𝑦𝜑)
7 dmopab3 5756 . . . . . 6 (∀𝑥 ∈ V ∃𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)} = V)
86, 7sylib 221 . . . . 5 (∃𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝜑)} = V)
94, 8syl5eq 2848 . . . 4 (∃𝑦𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = V)
10 vprc 5186 . . . . 5 ¬ V ∈ V
1110a1i 11 . . . 4 (∃𝑦𝜑 → ¬ V ∈ V)
129, 11eqneltrd 2912 . . 3 (∃𝑦𝜑 → ¬ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
13 dmexg 7598 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V → dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
1412, 13nsyl 142 . 2 (∃𝑦𝜑 → ¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
15 df-nel 3095 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∉ V ↔ ¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V)
1614, 15sylibr 237 1 (∃𝑦𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∉ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wex 1781  wcel 2112  wnel 3094  wral 3109  Vcvv 3444  {copab 5095  dom cdm 5523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-nel 3095  df-ral 3114  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-cnv 5531  df-dm 5533  df-rn 5534
This theorem is referenced by:  griedg0prc  27058
  Copyright terms: Public domain W3C validator