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Theorem eliuniincex 41809
 Description: Counterexample to show that the additional conditions in eliuniin 41799 and eliuniin2 41819 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 ¬ (𝑍𝐴 ↔ ∃𝑥𝐵𝑦𝐶 𝑍𝐷)
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
StepHypRef Expression
1 eliuniincex.4 . . 3 𝑍 = V
2 nvel 5185 . . 3 ¬ V ∈ 𝐴
31, 2eqneltri 2883 . 2 ¬ 𝑍𝐴
4 0ex 5176 . . . . 5 ∅ ∈ V
54snid 4561 . . . 4 ∅ ∈ {∅}
6 eliuniincex.1 . . . 4 𝐵 = {∅}
75, 6eleqtrri 2889 . . 3 ∅ ∈ 𝐵
8 ral0 4414 . . 3 𝑦 ∈ ∅ 𝑍𝐷
9 nfcv 2955 . . . . 5 𝑥
10 nfcv 2955 . . . . . 6 𝑥𝑍
11 eliuniincex.3 . . . . . . 7 𝐷 = ∅
1211, 9nfcxfr 2953 . . . . . 6 𝑥𝐷
1310, 12nfel 2969 . . . . 5 𝑥 𝑍𝐷
149, 13nfral 3190 . . . 4 𝑥𝑦 ∈ ∅ 𝑍𝐷
15 eliuniincex.2 . . . . . 6 𝐶 = ∅
1615raleqi 3362 . . . . 5 (∀𝑦𝐶 𝑍𝐷 ↔ ∀𝑦 ∈ ∅ 𝑍𝐷)
1716a1i 11 . . . 4 (𝑥 = ∅ → (∀𝑦𝐶 𝑍𝐷 ↔ ∀𝑦 ∈ ∅ 𝑍𝐷))
1814, 17rspce 3560 . . 3 ((∅ ∈ 𝐵 ∧ ∀𝑦 ∈ ∅ 𝑍𝐷) → ∃𝑥𝐵𝑦𝐶 𝑍𝐷)
197, 8, 18mp2an 691 . 2 𝑥𝐵𝑦𝐶 𝑍𝐷
20 pm3.22 463 . . . 4 ((¬ 𝑍𝐴 ∧ ∃𝑥𝐵𝑦𝐶 𝑍𝐷) → (∃𝑥𝐵𝑦𝐶 𝑍𝐷 ∧ ¬ 𝑍𝐴))
2120olcd 871 . . 3 ((¬ 𝑍𝐴 ∧ ∃𝑥𝐵𝑦𝐶 𝑍𝐷) → ((𝑍𝐴 ∧ ¬ ∃𝑥𝐵𝑦𝐶 𝑍𝐷) ∨ (∃𝑥𝐵𝑦𝐶 𝑍𝐷 ∧ ¬ 𝑍𝐴)))
22 xor 1012 . . 3 (¬ (𝑍𝐴 ↔ ∃𝑥𝐵𝑦𝐶 𝑍𝐷) ↔ ((𝑍𝐴 ∧ ¬ ∃𝑥𝐵𝑦𝐶 𝑍𝐷) ∨ (∃𝑥𝐵𝑦𝐶 𝑍𝐷 ∧ ¬ 𝑍𝐴)))
2321, 22sylibr 237 . 2 ((¬ 𝑍𝐴 ∧ ∃𝑥𝐵𝑦𝐶 𝑍𝐷) → ¬ (𝑍𝐴 ↔ ∃𝑥𝐵𝑦𝐶 𝑍𝐷))
243, 19, 23mp2an 691 1 ¬ (𝑍𝐴 ↔ ∃𝑥𝐵𝑦𝐶 𝑍𝐷)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  Vcvv 3441  ∅c0 4243  {csn 4525 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379  ax-ext 2770  ax-sep 5168  ax-nul 5175 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-nul 4244  df-sn 4526 This theorem is referenced by: (None)
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