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Theorem opab0 5235
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 5234 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝑦𝜑)
2 df-ne 3000 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ≠ ∅ ↔ ¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅)
3 2exnaln 1927 . . 3 (∃𝑥𝑦𝜑 ↔ ¬ ∀𝑥𝑦 ¬ 𝜑)
41, 2, 33bitr3i 293 . 2 (¬ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ ↔ ¬ ∀𝑥𝑦 ¬ 𝜑)
54con4bii 313 1 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ ↔ ∀𝑥𝑦 ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 198  wal 1654   = wceq 1656  wex 1878  wne 2999  c0 4146  {copab 4937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-opab 4938
This theorem is referenced by:  fvmptopab  6962  epinid0  8781  cnvepnep  8787  opabf  34677  sprsymrelfvlem  42605
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