Theorem opab0 5529

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 5528	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ≠ ∅ ↔ ∃𝑥∃𝑦𝜑)
2		df-ne 2961	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ≠ ∅ ↔ ¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅)
3		2exnaln 1852	. . 3 ⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑)
4	1, 2, 3	3bitr3i 304	. 2 ⊢ (¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑)
5	4	con4bii 324	1 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ ↔ ∀𝑥∀𝑦 ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 209  ∀wal 1561   = wceq 1563  ∃wex 1802   ≠ wne 2960  ∅c0 4288  {copab 5166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2737  ax-sep 5250  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-sn 4586  df-pr 4588  df-op 4592  df-opab 5167

This theorem is referenced by:  iresn0n0  6046  epinid0  9555  cnvepnep  9565  opabf  38882  tfsconcatb0  43928  sprsymrelfvlem  48095