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Theorem rabnc 3250
 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 3209 . 2 ({𝑥𝐴𝜑} ∩ {𝑥𝐴 ∣ ¬ 𝜑}) = {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜑)}
2 rabeq0 3247 . . 3 ({𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} = ∅ ↔ ∀𝑥𝐴 ¬ (𝜑 ∧ ¬ 𝜑))
3 pm3.24 627 . . . 4 ¬ (𝜑 ∧ ¬ 𝜑)
43a1i 9 . . 3 (𝑥𝐴 → ¬ (𝜑 ∧ ¬ 𝜑))
52, 4mprgbir 2379 . 2 {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜑)} = ∅
61, 5eqtri 2060 1 ({𝑥𝐴𝜑} ∩ {𝑥𝐴 ∣ ¬ 𝜑}) = ∅
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 97   = wceq 1243   ∈ wcel 1393  {crab 2310   ∩ cin 2916  ∅c0 3224 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-dif 2920  df-in 2924  df-nul 3225 This theorem is referenced by: (None)
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