Theorem eliincex 45147

Description: Counterexample to show that the additional conditions in eliin 4941 and eliin2 45153 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 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem eliincex

Step	Hyp	Ref	Expression
1		eliinct.1	. . 3 ⊢ 𝐴 = V
2		nvel 5249	. . 3 ⊢ ¬ V ∈ ∩ 𝑥 ∈ 𝐵 𝐶
3	1, 2	eqneltri 2850	. 2 ⊢ ¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶
4		ral0 4458	. . 3 ⊢ ∀𝑥 ∈ ∅ 𝐴 ∈ 𝐶
5		eliinct.2	. . . 4 ⊢ 𝐵 = ∅
6	5	raleqi 3290	. . 3 ⊢ (∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ↔ ∀𝑥 ∈ ∅ 𝐴 ∈ 𝐶)
7	4, 6	mpbir 231	. 2 ⊢ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶
8		pm3.22 459	. . . 4 ⊢ ((¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) → (∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ∧ ¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶))
9	8	olcd 874	. . 3 ⊢ ((¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) → ((𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ ¬ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) ∨ (∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ∧ ¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶)))
10		xor 1016	. . 3 ⊢ (¬ (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) ↔ ((𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ ¬ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) ∨ (∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ∧ ¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶)))
11	9, 10	sylibr 234	. 2 ⊢ ((¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) → ¬ (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
12	3, 7, 11	mp2an 692	1 ⊢ ¬ (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436  ∅c0 4278  ∩ ciin 4937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5229

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-v 3438  df-dif 3900  df-nul 4279

This theorem is referenced by: (None)