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Theorem 0nelopab 5525
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 Gino Giotto, 3-Oct-2024.)
Assertion
Ref Expression
0nelopab ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Proof of Theorem 0nelopab
StepHypRef Expression
1 vex 3450 . . . . . . 7 𝑥 ∈ V
2 vex 3450 . . . . . . 7 𝑦 ∈ V
31, 2opnzi 5432 . . . . . 6 𝑥, 𝑦⟩ ≠ ∅
43nesymi 3002 . . . . 5 ¬ ∅ = ⟨𝑥, 𝑦
54intnanr 489 . . . 4 ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
65nex 1803 . . 3 ¬ ∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
76nex 1803 . 2 ¬ ∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
8 elopab 5485 . 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 397   = wceq 1542  wex 1782  wcel 2107  c0 4283  cop 4593  {copab 5168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-opab 5169
This theorem is referenced by:  brabv  5527  epelg  5539  satf0n0  33975  bj-0nelmpt  35590
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