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Theorem bj-0nelopab 36250
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 5823 (this is a very limited reordering).

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

Proof of Theorem bj-0nelopab
StepHypRef Expression
1 relopab 5823 . 2 Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2 0nelrel0 5735 . 2 (Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
31, 2ax-mp 5 1 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2104  c0 4321  {copab 5209  Rel wrel 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-opab 5210  df-xp 5681  df-rel 5682
This theorem is referenced by: (None)
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