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Mirrors > Home > MPE Home > Th. List > Mathboxes > eliuniincex | Structured version Visualization version GIF version |
Description: Counterexample to show that the additional conditions in eliuniin 43397 and eliuniin2 43418 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.) |
Ref | Expression |
---|---|
eliuniincex.1 | ⊢ 𝐵 = {∅} |
eliuniincex.2 | ⊢ 𝐶 = ∅ |
eliuniincex.3 | ⊢ 𝐷 = ∅ |
eliuniincex.4 | ⊢ 𝑍 = V |
Ref | Expression |
---|---|
eliuniincex | ⊢ ¬ (𝑍 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliuniincex.4 | . . 3 ⊢ 𝑍 = V | |
2 | nvel 5274 | . . 3 ⊢ ¬ V ∈ 𝐴 | |
3 | 1, 2 | eqneltri 2853 | . 2 ⊢ ¬ 𝑍 ∈ 𝐴 |
4 | 0ex 5265 | . . . . 5 ⊢ ∅ ∈ V | |
5 | 4 | snid 4623 | . . . 4 ⊢ ∅ ∈ {∅} |
6 | eliuniincex.1 | . . . 4 ⊢ 𝐵 = {∅} | |
7 | 5, 6 | eleqtrri 2833 | . . 3 ⊢ ∅ ∈ 𝐵 |
8 | ral0 4471 | . . 3 ⊢ ∀𝑦 ∈ ∅ 𝑍 ∈ 𝐷 | |
9 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑥∅ | |
10 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑥𝑍 | |
11 | eliuniincex.3 | . . . . . . 7 ⊢ 𝐷 = ∅ | |
12 | 11, 9 | nfcxfr 2902 | . . . . . 6 ⊢ Ⅎ𝑥𝐷 |
13 | 10, 12 | nfel 2918 | . . . . 5 ⊢ Ⅎ𝑥 𝑍 ∈ 𝐷 |
14 | 9, 13 | nfral 3346 | . . . 4 ⊢ Ⅎ𝑥∀𝑦 ∈ ∅ 𝑍 ∈ 𝐷 |
15 | eliuniincex.2 | . . . . . 6 ⊢ 𝐶 = ∅ | |
16 | 15 | raleqi 3310 | . . . . 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 3569 | . . 3 ⊢ ((∅ ∈ 𝐵 ∧ ∀𝑦 ∈ ∅ 𝑍 ∈ 𝐷) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) |
19 | 7, 8, 18 | mp2an 691 | . 2 ⊢ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 |
20 | pm3.22 461 | . . . 4 ⊢ ((¬ 𝑍 ∈ 𝐴 ∧ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 ∧ ¬ 𝑍 ∈ 𝐴)) | |
21 | 20 | olcd 873 | . . 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 1014 | . . 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 233 | . 2 ⊢ ((¬ 𝑍 ∈ 𝐴 ∧ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → ¬ (𝑍 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷)) |
24 | 3, 19, 23 | mp2an 691 | 1 ⊢ ¬ (𝑍 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ∃wrex 3070 Vcvv 3444 ∅c0 4283 {csn 4587 |
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-10 2138 ax-11 2155 ax-12 2172 ax-13 2371 ax-ext 2704 ax-sep 5257 ax-nul 5264 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-v 3446 df-dif 3914 df-nul 4284 df-sn 4588 |
This theorem is referenced by: (None) |
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