Theorem bj-0nelopab 35947

Description: The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.) (Proof shortened by BJ, 22-Jul-2023.)

TODO: move to the main section when one can reorder sections so that we can use relopab 5825 (this is a very limited reordering).

Assertion

Ref	Expression
bj-0nelopab	⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Proof of Theorem bj-0nelopab

Step	Hyp	Ref	Expression
1		relopab 5825	. 2 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		0nelrel0 5737	. 2 ⊢ (Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
3	1, 2	ax-mp 5	1 ⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Syntax hints:  ¬ wn 3   ∈ wcel 2107  ∅c0 4323  {copab 5211  Rel wrel 5682

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-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

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 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-opab 5212  df-xp 5683  df-rel 5684

This theorem is referenced by: (None)