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Theorem 0nelopab 5577
Description: The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.) Reduce axiom usage and shorten proof. (Revised by GG, 3-Oct-2024.)
Assertion
Ref Expression
0nelopab ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Proof of Theorem 0nelopab
StepHypRef Expression
1 vex 3482 . . . . . . 7 𝑥 ∈ V
2 vex 3482 . . . . . . 7 𝑦 ∈ V
31, 2opnzi 5485 . . . . . 6 𝑥, 𝑦⟩ ≠ ∅
43nesymi 2996 . . . . 5 ¬ ∅ = ⟨𝑥, 𝑦
54intnanr 487 . . . 4 ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
65nex 1797 . . 3 ¬ ∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
76nex 1797 . 2 ¬ ∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
8 elopab 5537 . 2 (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
97, 8mtbir 323 1 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1537  wex 1776  wcel 2106  c0 4339  cop 4637  {copab 5210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211
This theorem is referenced by:  brabv  5578  epelg  5590  satf0n0  35363  bj-0nelmpt  37099
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