Theorem eliincex 45649

Description: Counterexample to show that the additional conditions in eliin 4951 and eliin2 45655 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 5266	. . 3 ⊢ ¬ V ∈ ∩ 𝑥 ∈ 𝐵 𝐶
3	1, 2	eqneltri 2880	. 2 ⊢ ¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶
4		ral0 4449	. . 3 ⊢ ∀𝑥 ∈ ∅ 𝐴 ∈ 𝐶
5		eliinct.2	. . . 4 ⊢ 𝐵 = ∅
6	5	raleqi 3317	. . 3 ⊢ (∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ↔ ∀𝑥 ∈ ∅ 𝐴 ∈ 𝐶)
7	4, 6	mpbir 233	. 2 ⊢ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶
8		pm3.22 463	. . . 4 ⊢ ((¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) → (∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ∧ ¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶))
9	8	olcd 885	. . 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 1027	. . 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 236	. 2 ⊢ ((¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) → ¬ (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
12	3, 7, 11	mp2an 702	1 ⊢ ¬ (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 399   ∨ wo 858   = wceq 1559   ∈ wcel 2141  ∀wral 3075  Vcvv 3453  ∅c0 4283  ∩ ciin 4947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-v 3455  df-dif 3905  df-nul 4284

This theorem is referenced by: (None)