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Theorem eliincex 44100
Description: Counterexample to show that the additional conditions in eliin 5001 and eliin2 44106 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 5315 . . 3 ¬ V ∈ 𝑥𝐵 𝐶
31, 2eqneltri 2850 . 2 ¬ 𝐴 𝑥𝐵 𝐶
4 ral0 4511 . . 3 𝑥 ∈ ∅ 𝐴𝐶
5 eliinct.2 . . . 4 𝐵 = ∅
65raleqi 3321 . . 3 (∀𝑥𝐵 𝐴𝐶 ↔ ∀𝑥 ∈ ∅ 𝐴𝐶)
74, 6mpbir 230 . 2 𝑥𝐵 𝐴𝐶
8 pm3.22 458 . . . 4 ((¬ 𝐴 𝑥𝐵 𝐶 ∧ ∀𝑥𝐵 𝐴𝐶) → (∀𝑥𝐵 𝐴𝐶 ∧ ¬ 𝐴 𝑥𝐵 𝐶))
98olcd 870 . . 3 ((¬ 𝐴 𝑥𝐵 𝐶 ∧ ∀𝑥𝐵 𝐴𝐶) → ((𝐴 𝑥𝐵 𝐶 ∧ ¬ ∀𝑥𝐵 𝐴𝐶) ∨ (∀𝑥𝐵 𝐴𝐶 ∧ ¬ 𝐴 𝑥𝐵 𝐶)))
10 xor 1011 . . 3 (¬ (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶) ↔ ((𝐴 𝑥𝐵 𝐶 ∧ ¬ ∀𝑥𝐵 𝐴𝐶) ∨ (∀𝑥𝐵 𝐴𝐶 ∧ ¬ 𝐴 𝑥𝐵 𝐶)))
119, 10sylibr 233 . 2 ((¬ 𝐴 𝑥𝐵 𝐶 ∧ ∀𝑥𝐵 𝐴𝐶) → ¬ (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
123, 7, 11mp2an 688 1 ¬ (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 394  wo 843   = wceq 1539  wcel 2104  wral 3059  Vcvv 3472  c0 4321   ciin 4997
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-v 3474  df-dif 3950  df-nul 4322
This theorem is referenced by: (None)
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