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Theorem rabnc 4391
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 4316 . 2 ({𝑥𝐴𝜑} ∩ {𝑥𝐴 ∣ ¬ 𝜑}) = {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜑)}
2 pm3.24 402 . . . 4 ¬ (𝜑 ∧ ¬ 𝜑)
32rgenw 3065 . . 3 𝑥𝐴 ¬ (𝜑 ∧ ¬ 𝜑)
4 rabeq0 4388 . . 3 ({𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} = ∅ ↔ ∀𝑥𝐴 ¬ (𝜑 ∧ ¬ 𝜑))
53, 4mpbir 231 . 2 {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} = ∅
61, 5eqtri 2765 1 ({𝑥𝐴𝜑} ∩ {𝑥𝐴 ∣ ¬ 𝜑}) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1540  wral 3061  {crab 3436  cin 3950  c0 4333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-in 3958  df-nul 4334
This theorem is referenced by:  elneldisj  4392  vtxdgoddnumeven  29571  esumrnmpt2  34069  hasheuni  34086  ddemeas  34237  ballotth  34540  jm2.22  43007
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