Step | Hyp | Ref
| Expression |
1 | | eleq1 2689 |
. . . . . . 7
⊢ (𝑣 = 𝑤 → (𝑣 ∈ 𝑥 ↔ 𝑤 ∈ 𝑥)) |
2 | | neeq2 2857 |
. . . . . . 7
⊢ (𝑣 = 𝑤 → (𝑧 ≠ 𝑣 ↔ 𝑧 ≠ 𝑤)) |
3 | 1, 2 | anbi12d 747 |
. . . . . 6
⊢ (𝑣 = 𝑤 → ((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) ↔ (𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤))) |
4 | | elequ2 2004 |
. . . . . . 7
⊢ (𝑣 = 𝑤 → (𝑦 ∈ 𝑣 ↔ 𝑦 ∈ 𝑤)) |
5 | 4 | notbid 308 |
. . . . . 6
⊢ (𝑣 = 𝑤 → (¬ 𝑦 ∈ 𝑣 ↔ ¬ 𝑦 ∈ 𝑤)) |
6 | 3, 5 | imbi12d 334 |
. . . . 5
⊢ (𝑣 = 𝑤 → (((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣) ↔ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ¬ 𝑦 ∈ 𝑤))) |
7 | 6 | spv 2260 |
. . . 4
⊢
(∀𝑣((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣) → ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ¬ 𝑦 ∈ 𝑤)) |
8 | | eldif 3584 |
. . . . 5
⊢ (𝑦 ∈ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ↔ (𝑦 ∈ 𝑧 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∖ {𝑧}))) |
9 | | simpr 477 |
. . . . . 6
⊢ ((𝑦 ∈ 𝑧 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∖ {𝑧})) → ¬ 𝑦 ∈ ∪ (𝑥 ∖ {𝑧})) |
10 | | eluni 4439 |
. . . . . . . 8
⊢ (𝑦 ∈ ∪ (𝑥
∖ {𝑧}) ↔
∃𝑣(𝑦 ∈ 𝑣 ∧ 𝑣 ∈ (𝑥 ∖ {𝑧}))) |
11 | 10 | notbii 310 |
. . . . . . 7
⊢ (¬
𝑦 ∈ ∪ (𝑥
∖ {𝑧}) ↔ ¬
∃𝑣(𝑦 ∈ 𝑣 ∧ 𝑣 ∈ (𝑥 ∖ {𝑧}))) |
12 | | alnex 1706 |
. . . . . . 7
⊢
(∀𝑣 ¬
(𝑦 ∈ 𝑣 ∧ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ¬ ∃𝑣(𝑦 ∈ 𝑣 ∧ 𝑣 ∈ (𝑥 ∖ {𝑧}))) |
13 | | con2b 349 |
. . . . . . . . 9
⊢ ((𝑦 ∈ 𝑣 → ¬ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ (𝑣 ∈ (𝑥 ∖ {𝑧}) → ¬ 𝑦 ∈ 𝑣)) |
14 | | imnan 438 |
. . . . . . . . 9
⊢ ((𝑦 ∈ 𝑣 → ¬ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ¬ (𝑦 ∈ 𝑣 ∧ 𝑣 ∈ (𝑥 ∖ {𝑧}))) |
15 | | eldifsn 4317 |
. . . . . . . . . . 11
⊢ (𝑣 ∈ (𝑥 ∖ {𝑧}) ↔ (𝑣 ∈ 𝑥 ∧ 𝑣 ≠ 𝑧)) |
16 | | necom 2847 |
. . . . . . . . . . . 12
⊢ (𝑣 ≠ 𝑧 ↔ 𝑧 ≠ 𝑣) |
17 | 16 | anbi2i 730 |
. . . . . . . . . . 11
⊢ ((𝑣 ∈ 𝑥 ∧ 𝑣 ≠ 𝑧) ↔ (𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣)) |
18 | 15, 17 | bitri 264 |
. . . . . . . . . 10
⊢ (𝑣 ∈ (𝑥 ∖ {𝑧}) ↔ (𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣)) |
19 | 18 | imbi1i 339 |
. . . . . . . . 9
⊢ ((𝑣 ∈ (𝑥 ∖ {𝑧}) → ¬ 𝑦 ∈ 𝑣) ↔ ((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣)) |
20 | 13, 14, 19 | 3bitr3i 290 |
. . . . . . . 8
⊢ (¬
(𝑦 ∈ 𝑣 ∧ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣)) |
21 | 20 | albii 1747 |
. . . . . . 7
⊢
(∀𝑣 ¬
(𝑦 ∈ 𝑣 ∧ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ∀𝑣((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣)) |
22 | 11, 12, 21 | 3bitr2i 288 |
. . . . . 6
⊢ (¬
𝑦 ∈ ∪ (𝑥
∖ {𝑧}) ↔
∀𝑣((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣)) |
23 | 9, 22 | sylib 208 |
. . . . 5
⊢ ((𝑦 ∈ 𝑧 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∖ {𝑧})) → ∀𝑣((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣)) |
24 | 8, 23 | sylbi 207 |
. . . 4
⊢ (𝑦 ∈ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) → ∀𝑣((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣)) |
25 | 7, 24 | syl11 33 |
. . 3
⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → (𝑦 ∈ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) → ¬ 𝑦 ∈ 𝑤)) |
26 | 25 | ralrimiv 2965 |
. 2
⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ∀𝑦 ∈ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ¬ 𝑦 ∈ 𝑤) |
27 | | disj 4017 |
. 2
⊢ (((𝑧 ∖ ∪ (𝑥
∖ {𝑧})) ∩ 𝑤) = ∅ ↔ ∀𝑦 ∈ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ¬ 𝑦 ∈ 𝑤) |
28 | 26, 27 | sylibr 224 |
1
⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅) |