Theorem eliuniincex 41373

Description: Counterexample to show that the additional conditions in eliuniin 41363 and eliuniin2 41384 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 5219	. . 3 ⊢ ¬ V ∈ 𝐴
3	1, 2	eqneltri 2906	. 2 ⊢ ¬ 𝑍 ∈ 𝐴
4		0ex 5210	. . . . 5 ⊢ ∅ ∈ V
5	4	snid 4600	. . . 4 ⊢ ∅ ∈ {∅}
6		eliuniincex.1	. . . 4 ⊢ 𝐵 = {∅}
7	5, 6	eleqtrri 2912	. . 3 ⊢ ∅ ∈ 𝐵
8		ral0 4455	. . 3 ⊢ ∀𝑦 ∈ ∅ 𝑍 ∈ 𝐷
9		nfcv 2977	. . . . 5 ⊢ Ⅎ𝑥∅
10		nfcv 2977	. . . . . 6 ⊢ Ⅎ𝑥𝑍
11		eliuniincex.3	. . . . . . 7 ⊢ 𝐷 = ∅
12	11, 9	nfcxfr 2975	. . . . . 6 ⊢ Ⅎ𝑥𝐷
13	10, 12	nfel 2992	. . . . 5 ⊢ Ⅎ𝑥 𝑍 ∈ 𝐷
14	9, 13	nfral 3226	. . . 4 ⊢ Ⅎ𝑥∀𝑦 ∈ ∅ 𝑍 ∈ 𝐷
15		eliuniincex.2	. . . . . 6 ⊢ 𝐶 = ∅
16	15	raleqi 3413	. . . . 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 690	. 2 ⊢ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷
20		pm3.22 462	. . . 4 ⊢ ((¬ 𝑍 ∈ 𝐴 ∧ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 ∧ ¬ 𝑍 ∈ 𝐴))
21	20	olcd 870	. . 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 1011	. . 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 236	. 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 208   ∧ wa 398   ∨ wo 843   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139  Vcvv 3494  ∅c0 4290  {csn 4566

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-13 2386  ax-ext 2793  ax-sep 5202  ax-nul 5209

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-v 3496  df-dif 3938  df-nul 4291  df-sn 4567

This theorem is referenced by: (None)