Theorem eliuniincex 45049

Description: Counterexample to show that the additional conditions in eliuniin 45039 and eliuniin2 45060 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 5322	. . 3 ⊢ ¬ V ∈ 𝐴
3	1, 2	eqneltri 2858	. 2 ⊢ ¬ 𝑍 ∈ 𝐴
4		0ex 5313	. . . . 5 ⊢ ∅ ∈ V
5	4	snid 4667	. . . 4 ⊢ ∅ ∈ {∅}
6		eliuniincex.1	. . . 4 ⊢ 𝐵 = {∅}
7	5, 6	eleqtrri 2838	. . 3 ⊢ ∅ ∈ 𝐵
8		ral0 4519	. . 3 ⊢ ∀𝑦 ∈ ∅ 𝑍 ∈ 𝐷
9		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑥∅
10		nfcv 2903	. . . . . 6 ⊢ Ⅎ𝑥𝑍
11		eliuniincex.3	. . . . . . 7 ⊢ 𝐷 = ∅
12	11, 9	nfcxfr 2901	. . . . . 6 ⊢ Ⅎ𝑥𝐷
13	10, 12	nfel 2918	. . . . 5 ⊢ Ⅎ𝑥 𝑍 ∈ 𝐷
14	9, 13	nfral 3372	. . . 4 ⊢ Ⅎ𝑥∀𝑦 ∈ ∅ 𝑍 ∈ 𝐷
15		eliuniincex.2	. . . . . 6 ⊢ 𝐶 = ∅
16	15	raleqi 3322	. . . . 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 3611	. . 3 ⊢ ((∅ ∈ 𝐵 ∧ ∀𝑦 ∈ ∅ 𝑍 ∈ 𝐷) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷)
19	7, 8, 18	mp2an 692	. 2 ⊢ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷
20		pm3.22 459	. . . 4 ⊢ ((¬ 𝑍 ∈ 𝐴 ∧ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 ∧ ¬ 𝑍 ∈ 𝐴))
21	20	olcd 874	. . 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 1016	. . 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 234	. 2 ⊢ ((¬ 𝑍 ∈ 𝐴 ∧ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → ¬ (𝑍 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷))
24	3, 19, 23	mp2an 692	1 ⊢ ¬ (𝑍 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷)

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068  Vcvv 3478  ∅c0 4339  {csn 4631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375  ax-ext 2706  ax-sep 5302  ax-nul 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-v 3480  df-dif 3966  df-nul 4340  df-sn 4632

This theorem is referenced by: (None)