Theorem eliincex 40103

Description: Counterexample to show that the additional conditions in eliin 4747 and eliin2 40109 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 5025	. . 3 ⊢ ¬ V ∈ ∩ 𝑥 ∈ 𝐵 𝐶
3	1, 2	eqneltri 40058	. 2 ⊢ ¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶
4		ral0 4300	. . 3 ⊢ ∀𝑥 ∈ ∅ 𝐴 ∈ 𝐶
5		eliinct.2	. . . 4 ⊢ 𝐵 = ∅
6	5	raleqi 3354	. . 3 ⊢ (∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ↔ ∀𝑥 ∈ ∅ 𝐴 ∈ 𝐶)
7	4, 6	mpbir 223	. 2 ⊢ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶
8		pm3.22 453	. . . 4 ⊢ ((¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) → (∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ∧ ¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶))
9	8	olcd 905	. . 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 1043	. . 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 226	. 2 ⊢ ((¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) → ¬ (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
12	3, 7, 11	mp2an 683	1 ⊢ ¬ (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)

Syntax hints:  ¬ wn 3   ↔ wb 198   ∧ wa 386   ∨ wo 878   = wceq 1656   ∈ wcel 2164  ∀wral 3117  Vcvv 3414  ∅c0 4146  ∩ ciin 4743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-v 3416  df-dif 3801  df-nul 4147

This theorem is referenced by: (None)