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 2934	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ≠ ∅ ↔ ¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅)
3		2exnaln 1824	. . 3 ⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑)
4	1, 2, 3	3bitr3i 300	. 2 ⊢ (¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑)
5	4	con4bii 320	1 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ ↔ ∀𝑥∀𝑦 ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 205  ∀wal 1532   = wceq 1534  ∃wex 1774   ≠ wne 2933  ∅c0 4335  {copab 5218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-11 2150  ax-ext 2700  ax-sep 5307  ax-nul 5314  ax-pr 5437

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2062  df-clab 2707  df-cleq 2721  df-clel 2806  df-ne 2934  df-v 3474  df-dif 3962  df-un 3964  df-ss 3976  df-nul 4336  df-if 4537  df-sn 4637  df-pr 4639  df-op 4643  df-opab 5219

This theorem is referenced by:  iresn0n0  6068  fvmptopabOLD  7486  epinid0  9650  cnvepnep  9658  opabf  38133  tfsconcatb0  43101  sprsymrelfvlem  47152