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Theorem opab0 5428
Description: Empty ordered pair class abstraction. (Contributed by AV, 29-Oct-2021.)
Assertion
Ref Expression
opab0 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ ↔ ∀𝑥𝑦 ¬ 𝜑)

Proof of Theorem opab0
StepHypRef Expression
1 opabn0 5427 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝑦𝜑)
2 df-ne 3015 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ≠ ∅ ↔ ¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅)
3 2exnaln 1830 . . 3 (∃𝑥𝑦𝜑 ↔ ¬ ∀𝑥𝑦 ¬ 𝜑)
41, 2, 33bitr3i 304 . 2 (¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ ↔ ¬ ∀𝑥𝑦 ¬ 𝜑)
54con4bii 324 1 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ ↔ ∀𝑥𝑦 ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wal 1536   = wceq 1538  wex 1781  wne 3014  c0 4276  {copab 5114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-opab 5115
This theorem is referenced by:  iresn0n0  5910  fvmptopab  7202  epinid0  9061  cnvepnep  9068  opabf  35728  sprsymrelfvlem  43937
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