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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 5001 and eliin2 45056 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 5322 | . . 3 ⊢ ¬ V ∈ ∩ 𝑥 ∈ 𝐵 𝐶 | |
3 | 1, 2 | eqneltri 2858 | . 2 ⊢ ¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 |
4 | ral0 4519 | . . 3 ⊢ ∀𝑥 ∈ ∅ 𝐴 ∈ 𝐶 | |
5 | eliinct.2 | . . . 4 ⊢ 𝐵 = ∅ | |
6 | 5 | raleqi 3322 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ↔ ∀𝑥 ∈ ∅ 𝐴 ∈ 𝐶) |
7 | 4, 6 | mpbir 231 | . 2 ⊢ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 |
8 | pm3.22 459 | . . . 4 ⊢ ((¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) → (∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ∧ ¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶)) | |
9 | 8 | olcd 874 | . . 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 1016 | . . 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 234 | . 2 ⊢ ((¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) → ¬ (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)) |
12 | 3, 7, 11 | mp2an 692 | 1 ⊢ ¬ (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 ∅c0 4339 ∩ ciin 4997 |
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-ext 2706 ax-sep 5302 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-v 3480 df-dif 3966 df-nul 4340 |
This theorem is referenced by: (None) |
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