Theorem 0nelopab 5547

Description: The empty set is not an element of 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

Step	Hyp	Ref	Expression
1		vex 3458	. . . . . . 7 ⊢ 𝑥 ∈ V
2		vex 3458	. . . . . . 7 ⊢ 𝑦 ∈ V
3	1, 2	opnzi 5450	. . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ ≠ ∅
4	3	nesymi 3016	. . . . 5 ⊢ ¬ ∅ = ⟨𝑥, 𝑦⟩
5	4	intnanr 493	. . . 4 ⊢ ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
6	5	nex 1833	. . 3 ⊢ ¬ ∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
7	6	nex 1833	. 2 ⊢ ¬ ∃𝑥∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
8		elopab 5508	. 2 ⊢ (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
9	7, 8	mtbir 326	1 ⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∅c0 4281  ⟨cop 4840  {copab 5173

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-opab 5174

This theorem is used by:  brabv  5548  epelg  5559  satf0n0  36351  bj-0nelmpt  38308