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Theorem rabnc 3916
 Description: Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
Assertion
Ref Expression
rabnc ({𝑥𝐴𝜑} ∩ {𝑥𝐴 ∣ ¬ 𝜑}) = ∅
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabnc
StepHypRef Expression
1 inrab 3858 . 2 ({𝑥𝐴𝜑} ∩ {𝑥𝐴 ∣ ¬ 𝜑}) = {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜑)}
2 pm3.24 922 . . . 4 ¬ (𝜑 ∧ ¬ 𝜑)
32rgenw 2908 . . 3 𝑥𝐴 ¬ (𝜑 ∧ ¬ 𝜑)
4 rabeq0 3911 . . 3 ({𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} = ∅ ↔ ∀𝑥𝐴 ¬ (𝜑 ∧ ¬ 𝜑))
53, 4mpbir 220 . 2 {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} = ∅
61, 5eqtri 2632 1 ({𝑥𝐴𝜑} ∩ {𝑥𝐴 ∣ ¬ 𝜑}) = ∅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 383   = wceq 1475  ∀wral 2896  {crab 2900   ∩ cin 3539  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-in 3547  df-nul 3875 This theorem is referenced by:  elneldisj  3917  esumrnmpt2  29457  hasheuni  29474  ddemeas  29626  ballotth  29926  jm2.22  36580
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