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Theorem 0nelopab 5477
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 3435 . . . . . . 7 𝑥 ∈ V
2 vex 3435 . . . . . . 7 𝑦 ∈ V
31, 2opnzi 5389 . . . . . 6 𝑥, 𝑦⟩ ≠ ∅
43nesymi 3001 . . . . 5 ¬ ∅ = ⟨𝑥, 𝑦
54intnanr 488 . . . 4 ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
65nex 1803 . . 3 ¬ ∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
76nex 1803 . 2 ¬ ∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
8 elopab 5439 . 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 396   = wceq 1539  wex 1782  wcel 2106  c0 4258  cop 4569  {copab 5137
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-v 3433  df-dif 3891  df-un 3893  df-nul 4259  df-if 4462  df-sn 4564  df-pr 4566  df-op 4570  df-opab 5138
This theorem is referenced by:  brabv  5479  epelg  5493  satf0n0  33327  bj-0nelmpt  35274
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