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Theorem eliincex 38324
Description: Counterexample to show that the additional conditions in eliin 4461 and eliin2 38330 are actually needed. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
eliinct.1 𝐴 = V
eliinct.2 𝐵 = ∅
Assertion
Ref Expression
eliincex ¬ (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem eliincex
StepHypRef Expression
1 eliinct.1 . . 3 𝐴 = V
2 nvel 4725 . . 3 ¬ V ∈ 𝑥𝐵 𝐶
31, 2eqneltri 38272 . 2 ¬ 𝐴 𝑥𝐵 𝐶
4 ral0 4028 . . 3 𝑥 ∈ ∅ 𝐴𝐶
5 eliinct.2 . . . 4 𝐵 = ∅
65raleqi 3119 . . 3 (∀𝑥𝐵 𝐴𝐶 ↔ ∀𝑥 ∈ ∅ 𝐴𝐶)
74, 6mpbir 220 . 2 𝑥𝐵 𝐴𝐶
8 pm3.22 464 . . . 4 ((¬ 𝐴 𝑥𝐵 𝐶 ∧ ∀𝑥𝐵 𝐴𝐶) → (∀𝑥𝐵 𝐴𝐶 ∧ ¬ 𝐴 𝑥𝐵 𝐶))
98olcd 407 . . 3 ((¬ 𝐴 𝑥𝐵 𝐶 ∧ ∀𝑥𝐵 𝐴𝐶) → ((𝐴 𝑥𝐵 𝐶 ∧ ¬ ∀𝑥𝐵 𝐴𝐶) ∨ (∀𝑥𝐵 𝐴𝐶 ∧ ¬ 𝐴 𝑥𝐵 𝐶)))
10 xor 931 . . 3 (¬ (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶) ↔ ((𝐴 𝑥𝐵 𝐶 ∧ ¬ ∀𝑥𝐵 𝐴𝐶) ∨ (∀𝑥𝐵 𝐴𝐶 ∧ ¬ 𝐴 𝑥𝐵 𝐶)))
119, 10sylibr 223 . 2 ((¬ 𝐴 𝑥𝐵 𝐶 ∧ ∀𝑥𝐵 𝐴𝐶) → ¬ (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
123, 7, 11mp2an 704 1 ¬ (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 195  wo 382  wa 383   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173  c0 3874   ciin 4456
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-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709
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-v 3175  df-dif 3543  df-nul 3875
This theorem is referenced by: (None)
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