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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > eliincex | Structured version Visualization version GIF version |
Description: Counterexample to show that the additional conditions in eliin 5003 and eliin2 43805 are actually needed. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
eliinct.1 | ⊢ 𝐴 = V |
eliinct.2 | ⊢ 𝐵 = ∅ |
Ref | Expression |
---|---|
eliincex | ⊢ ¬ (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliinct.1 | . . 3 ⊢ 𝐴 = V | |
2 | nvel 5317 | . . 3 ⊢ ¬ V ∈ ∩ 𝑥 ∈ 𝐵 𝐶 | |
3 | 1, 2 | eqneltri 2853 | . 2 ⊢ ¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 |
4 | ral0 4513 | . . 3 ⊢ ∀𝑥 ∈ ∅ 𝐴 ∈ 𝐶 | |
5 | eliinct.2 | . . . 4 ⊢ 𝐵 = ∅ | |
6 | 5 | raleqi 3324 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ↔ ∀𝑥 ∈ ∅ 𝐴 ∈ 𝐶) |
7 | 4, 6 | mpbir 230 | . 2 ⊢ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 |
8 | pm3.22 461 | . . . 4 ⊢ ((¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) → (∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ∧ ¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶)) | |
9 | 8 | olcd 873 | . . 3 ⊢ ((¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) → ((𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ ¬ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) ∨ (∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ∧ ¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶))) |
10 | xor 1014 | . . 3 ⊢ (¬ (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) ↔ ((𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ ¬ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) ∨ (∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ∧ ¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶))) | |
11 | 9, 10 | sylibr 233 | . 2 ⊢ ((¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) → ¬ (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)) |
12 | 3, 7, 11 | mp2an 691 | 1 ⊢ ¬ (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ∀wral 3062 Vcvv 3475 ∅c0 4323 ∩ ciin 4999 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-v 3477 df-dif 3952 df-nul 4324 |
This theorem is referenced by: (None) |
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