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Theorem eliuniincex 39809
Description: Counterexample to show that the additional conditions in eliuniin 39796 and eliuniin2 39820 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 4949 . . 3 ¬ V ∈ 𝐴
31, 2eqneltri 39763 . 2 ¬ 𝑍𝐴
4 0ex 4942 . . . . 5 ∅ ∈ V
54snid 4353 . . . 4 ∅ ∈ {∅}
6 eliuniincex.1 . . . 4 𝐵 = {∅}
75, 6eleqtrri 2838 . . 3 ∅ ∈ 𝐵
8 ral0 4220 . . 3 𝑦 ∈ ∅ 𝑍𝐷
9 nfcv 2902 . . . . 5 𝑥
10 nfcv 2902 . . . . . 6 𝑥𝑍
11 eliuniincex.3 . . . . . . 7 𝐷 = ∅
1211, 9nfcxfr 2900 . . . . . 6 𝑥𝐷
1310, 12nfel 2915 . . . . 5 𝑥 𝑍𝐷
149, 13nfral 3083 . . . 4 𝑥𝑦 ∈ ∅ 𝑍𝐷
15 eliuniincex.2 . . . . . 6 𝐶 = ∅
1615raleqi 3281 . . . . 5 (∀𝑦𝐶 𝑍𝐷 ↔ ∀𝑦 ∈ ∅ 𝑍𝐷)
1716a1i 11 . . . 4 (𝑥 = ∅ → (∀𝑦𝐶 𝑍𝐷 ↔ ∀𝑦 ∈ ∅ 𝑍𝐷))
1814, 17rspce 3444 . . 3 ((∅ ∈ 𝐵 ∧ ∀𝑦 ∈ ∅ 𝑍𝐷) → ∃𝑥𝐵𝑦𝐶 𝑍𝐷)
197, 8, 18mp2an 710 . 2 𝑥𝐵𝑦𝐶 𝑍𝐷
20 pm3.22 464 . . . 4 ((¬ 𝑍𝐴 ∧ ∃𝑥𝐵𝑦𝐶 𝑍𝐷) → (∃𝑥𝐵𝑦𝐶 𝑍𝐷 ∧ ¬ 𝑍𝐴))
2120olcd 407 . . 3 ((¬ 𝑍𝐴 ∧ ∃𝑥𝐵𝑦𝐶 𝑍𝐷) → ((𝑍𝐴 ∧ ¬ ∃𝑥𝐵𝑦𝐶 𝑍𝐷) ∨ (∃𝑥𝐵𝑦𝐶 𝑍𝐷 ∧ ¬ 𝑍𝐴)))
22 xor 971 . . 3 (¬ (𝑍𝐴 ↔ ∃𝑥𝐵𝑦𝐶 𝑍𝐷) ↔ ((𝑍𝐴 ∧ ¬ ∃𝑥𝐵𝑦𝐶 𝑍𝐷) ∨ (∃𝑥𝐵𝑦𝐶 𝑍𝐷 ∧ ¬ 𝑍𝐴)))
2321, 22sylibr 224 . 2 ((¬ 𝑍𝐴 ∧ ∃𝑥𝐵𝑦𝐶 𝑍𝐷) → ¬ (𝑍𝐴 ↔ ∃𝑥𝐵𝑦𝐶 𝑍𝐷))
243, 19, 23mp2an 710 1 ¬ (𝑍𝐴 ↔ ∃𝑥𝐵𝑦𝐶 𝑍𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wo 382  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  Vcvv 3340  c0 4058  {csn 4321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-dif 3718  df-nul 4059  df-sn 4322
This theorem is referenced by: (None)
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