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Theorem rabnc 4383
Description: Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
Assertion
Ref Expression
rabnc ({𝑥𝐴𝜑} ∩ {𝑥𝐴 ∣ ¬ 𝜑}) = ∅

Proof of Theorem rabnc
StepHypRef Expression
1 inrab 4302 . 2 ({𝑥𝐴𝜑} ∩ {𝑥𝐴 ∣ ¬ 𝜑}) = {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜑)}
2 pm3.24 402 . . . 4 ¬ (𝜑 ∧ ¬ 𝜑)
32rgenw 3061 . . 3 𝑥𝐴 ¬ (𝜑 ∧ ¬ 𝜑)
4 rabeq0 4380 . . 3 ({𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} = ∅ ↔ ∀𝑥𝐴 ¬ (𝜑 ∧ ¬ 𝜑))
53, 4mpbir 230 . 2 {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} = ∅
61, 5eqtri 2756 1 ({𝑥𝐴𝜑} ∩ {𝑥𝐴 ∣ ¬ 𝜑}) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1534  wral 3057  {crab 3428  cin 3944  c0 4318
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 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rab 3429  df-v 3472  df-dif 3948  df-in 3952  df-nul 4319
This theorem is referenced by:  elneldisj  4384  vtxdgoddnumeven  29360  esumrnmpt2  33681  hasheuni  33698  ddemeas  33849  ballotth  34151  jm2.22  42410
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