Theorem opab0 5564

Description: Empty ordered pair class abstraction. (Contributed by AV, 29-Oct-2021.)

Assertion

Ref	Expression
opab0	⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ ↔ ∀𝑥∀𝑦 ¬ 𝜑)

Proof of Theorem opab0

Step	Hyp	Ref	Expression
1		opabn0 5563	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ≠ ∅ ↔ ∃𝑥∃𝑦𝜑)
2		df-ne 2939	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ≠ ∅ ↔ ¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅)
3		2exnaln 1826	. . 3 ⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑)
4	1, 2, 3	3bitr3i 301	. 2 ⊢ (¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑)
5	4	con4bii 321	1 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ ↔ ∀𝑥∀𝑦 ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 206  ∀wal 1535   = wceq 1537  ∃wex 1776   ≠ wne 2938  ∅c0 4339  {copab 5210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211

This theorem is referenced by:  iresn0n0  6074  fvmptopabOLD  7488  epinid0  9638  cnvepnep  9646  opabf  38350  tfsconcatb0  43334  sprsymrelfvlem  47415