Theorem 0nelopab 5471

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

Step	Hyp	Ref	Expression
1		vex 3426	. . . . . . 7 ⊢ 𝑥 ∈ V
2		vex 3426	. . . . . . 7 ⊢ 𝑦 ∈ V
3	1, 2	opnzi 5383	. . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ ≠ ∅
4	3	nesymi 3000	. . . . 5 ⊢ ¬ ∅ = ⟨𝑥, 𝑦⟩
5	4	intnanr 487	. . . 4 ⊢ ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
6	5	nex 1804	. . 3 ⊢ ¬ ∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
7	6	nex 1804	. 2 ⊢ ¬ ∃𝑥∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
8		elopab 5433	. 2 ⊢ (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
9	7, 8	mtbir 322	1 ⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∅c0 4253  ⟨cop 4564  {copab 5132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-opab 5133

This theorem is referenced by:  brabv  5473  epelg  5487  satf0n0  33240  bj-0nelmpt  35214