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Theorem rabnc 4106
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 4047 . 2 ({𝑥𝐴𝜑} ∩ {𝑥𝐴 ∣ ¬ 𝜑}) = {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜑)}
2 pm3.24 389 . . . 4 ¬ (𝜑 ∧ ¬ 𝜑)
32rgenw 3073 . . 3 𝑥𝐴 ¬ (𝜑 ∧ ¬ 𝜑)
4 rabeq0 4103 . . 3 ({𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} = ∅ ↔ ∀𝑥𝐴 ¬ (𝜑 ∧ ¬ 𝜑))
53, 4mpbir 221 . 2 {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} = ∅
61, 5eqtri 2793 1 ({𝑥𝐴𝜑} ∩ {𝑥𝐴 ∣ ¬ 𝜑}) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 382   = wceq 1631  wral 3061  {crab 3065  cin 3722  c0 4063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-in 3730  df-nul 4064
This theorem is referenced by:  elneldisj  4107  elneldisjOLD  4109  vtxdgoddnumeven  26684  esumrnmpt2  30470  hasheuni  30487  ddemeas  30639  ballotth  30939  jm2.22  38088
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