Theorem eliuniincex 43441

Description: Counterexample to show that the additional conditions in eliuniin 43431 and eliuniin2 43452 are actually needed. Notice that the definition of 𝐴 is not even needed (it can be any class). (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
eliuniincex.1	⊢ 𝐵 = {∅}
eliuniincex.2	⊢ 𝐶 = ∅
eliuniincex.3	⊢ 𝐷 = ∅
eliuniincex.4	⊢ 𝑍 = V

Assertion

Ref	Expression
eliuniincex	⊢ ¬ (𝑍 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷)

Distinct variable groups:   𝑥,𝐵   𝑦,𝐶   𝑥,𝑍

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑦)   𝐶(𝑥)   𝐷(𝑥,𝑦)   𝑍(𝑦)

Proof of Theorem eliuniincex

Step	Hyp	Ref	Expression
1		eliuniincex.4	. . 3 ⊢ 𝑍 = V
2		nvel 5278	. . 3 ⊢ ¬ V ∈ 𝐴
3	1, 2	eqneltri 2851	. 2 ⊢ ¬ 𝑍 ∈ 𝐴
4		0ex 5269	. . . . 5 ⊢ ∅ ∈ V
5	4	snid 4627	. . . 4 ⊢ ∅ ∈ {∅}
6		eliuniincex.1	. . . 4 ⊢ 𝐵 = {∅}
7	5, 6	eleqtrri 2831	. . 3 ⊢ ∅ ∈ 𝐵
8		ral0 4475	. . 3 ⊢ ∀𝑦 ∈ ∅ 𝑍 ∈ 𝐷
9		nfcv 2902	. . . . 5 ⊢ Ⅎ𝑥∅
10		nfcv 2902	. . . . . 6 ⊢ Ⅎ𝑥𝑍
11		eliuniincex.3	. . . . . . 7 ⊢ 𝐷 = ∅
12	11, 9	nfcxfr 2900	. . . . . 6 ⊢ Ⅎ𝑥𝐷
13	10, 12	nfel 2916	. . . . 5 ⊢ Ⅎ𝑥 𝑍 ∈ 𝐷
14	9, 13	nfral 3345	. . . 4 ⊢ Ⅎ𝑥∀𝑦 ∈ ∅ 𝑍 ∈ 𝐷
15		eliuniincex.2	. . . . . 6 ⊢ 𝐶 = ∅
16	15	raleqi 3309	. . . . 5 ⊢ (∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 ↔ ∀𝑦 ∈ ∅ 𝑍 ∈ 𝐷)
17	16	a1i 11	. . . 4 ⊢ (𝑥 = ∅ → (∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 ↔ ∀𝑦 ∈ ∅ 𝑍 ∈ 𝐷))
18	14, 17	rspce 3571	. . 3 ⊢ ((∅ ∈ 𝐵 ∧ ∀𝑦 ∈ ∅ 𝑍 ∈ 𝐷) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷)
19	7, 8, 18	mp2an 690	. 2 ⊢ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷
20		pm3.22 460	. . . 4 ⊢ ((¬ 𝑍 ∈ 𝐴 ∧ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 ∧ ¬ 𝑍 ∈ 𝐴))
21	20	olcd 872	. . 3 ⊢ ((¬ 𝑍 ∈ 𝐴 ∧ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → ((𝑍 ∈ 𝐴 ∧ ¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) ∨ (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 ∧ ¬ 𝑍 ∈ 𝐴)))
22		xor 1013	. . 3 ⊢ (¬ (𝑍 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) ↔ ((𝑍 ∈ 𝐴 ∧ ¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) ∨ (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 ∧ ¬ 𝑍 ∈ 𝐴)))
23	21, 22	sylibr 233	. 2 ⊢ ((¬ 𝑍 ∈ 𝐴 ∧ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → ¬ (𝑍 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷))
24	3, 19, 23	mp2an 690	1 ⊢ ¬ (𝑍 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷)

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106  ∀wral 3060  ∃wrex 3069  Vcvv 3446  ∅c0 4287  {csn 4591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2370  ax-ext 2702  ax-sep 5261  ax-nul 5268

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-v 3448  df-dif 3916  df-nul 4288  df-sn 4592

This theorem is referenced by: (None)