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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 45104 and eliuniin2 45125 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 5316 | . . 3 ⊢ ¬ V ∈ 𝐴 | |
| 3 | 1, 2 | eqneltri 2860 | . 2 ⊢ ¬ 𝑍 ∈ 𝐴 |
| 4 | 0ex 5307 | . . . . 5 ⊢ ∅ ∈ V | |
| 5 | 4 | snid 4662 | . . . 4 ⊢ ∅ ∈ {∅} |
| 6 | eliuniincex.1 | . . . 4 ⊢ 𝐵 = {∅} | |
| 7 | 5, 6 | eleqtrri 2840 | . . 3 ⊢ ∅ ∈ 𝐵 |
| 8 | ral0 4513 | . . 3 ⊢ ∀𝑦 ∈ ∅ 𝑍 ∈ 𝐷 | |
| 9 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥∅ | |
| 10 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑥𝑍 | |
| 11 | eliuniincex.3 | . . . . . . 7 ⊢ 𝐷 = ∅ | |
| 12 | 11, 9 | nfcxfr 2903 | . . . . . 6 ⊢ Ⅎ𝑥𝐷 |
| 13 | 10, 12 | nfel 2920 | . . . . 5 ⊢ Ⅎ𝑥 𝑍 ∈ 𝐷 |
| 14 | 9, 13 | nfral 3374 | . . . 4 ⊢ Ⅎ𝑥∀𝑦 ∈ ∅ 𝑍 ∈ 𝐷 |
| 15 | eliuniincex.2 | . . . . . 6 ⊢ 𝐶 = ∅ | |
| 16 | 15 | raleqi 3324 | . . . . 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 3611 | . . 3 ⊢ ((∅ ∈ 𝐵 ∧ ∀𝑦 ∈ ∅ 𝑍 ∈ 𝐷) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) |
| 19 | 7, 8, 18 | mp2an 692 | . 2 ⊢ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 |
| 20 | pm3.22 459 | . . . 4 ⊢ ((¬ 𝑍 ∈ 𝐴 ∧ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 ∧ ¬ 𝑍 ∈ 𝐴)) | |
| 21 | 20 | olcd 875 | . . 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 1017 | . . 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 234 | . 2 ⊢ ((¬ 𝑍 ∈ 𝐴 ∧ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → ¬ (𝑍 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷)) |
| 24 | 3, 19, 23 | mp2an 692 | 1 ⊢ ¬ (𝑍 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ∅c0 4333 {csn 4626 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 ax-ext 2708 ax-sep 5296 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-v 3482 df-dif 3954 df-nul 4334 df-sn 4627 |
| This theorem is referenced by: (None) |
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