Step | Hyp | Ref
| Expression |
1 | | df-disj 3967 |
. . . 4
⊢
(Disj 𝑥
∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) |
2 | | elin 3310 |
. . . . . . . . . 10
⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪
𝑥 ∈ 𝐷 𝐵) ↔ (𝑦 ∈ ∪
𝑥 ∈ 𝐶 𝐵 ∧ 𝑦 ∈ ∪
𝑥 ∈ 𝐷 𝐵)) |
3 | | eliun 3877 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐶 𝐵 ↔ ∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵) |
4 | | eliun 3877 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐷 𝐵 ↔ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵) |
5 | 3, 4 | anbi12i 457 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐶 𝐵 ∧ 𝑦 ∈ ∪
𝑥 ∈ 𝐷 𝐵) ↔ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) |
6 | 2, 5 | bitri 183 |
. . . . . . . . 9
⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪
𝑥 ∈ 𝐷 𝐵) ↔ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) |
7 | | nfv 1521 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑤 𝑦 ∈ 𝐵 |
8 | 7 | rmo3 3046 |
. . . . . . . . . . 11
⊢
(∃*𝑥 ∈
𝐴 𝑦 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤)) |
9 | | simprl 526 |
. . . . . . . . . . . . . 14
⊢
(((∀𝑥 ∈
𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) → ∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵) |
10 | | nfv 1521 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑢 𝑦 ∈ 𝐵 |
11 | | nfcsb1v 3082 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑥⦋𝑢 / 𝑥⦌𝐵 |
12 | 11 | nfcri 2306 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑥 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵 |
13 | | csbeq1a 3058 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑢 → 𝐵 = ⦋𝑢 / 𝑥⦌𝐵) |
14 | 13 | eleq2d 2240 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑢 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) |
15 | 10, 12, 14 | cbvrex 2693 |
. . . . . . . . . . . . . 14
⊢
(∃𝑥 ∈
𝐶 𝑦 ∈ 𝐵 ↔ ∃𝑢 ∈ 𝐶 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵) |
16 | 9, 15 | sylib 121 |
. . . . . . . . . . . . 13
⊢
(((∀𝑥 ∈
𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) → ∃𝑢 ∈ 𝐶 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵) |
17 | | simplrr 531 |
. . . . . . . . . . . . . . 15
⊢
((((∀𝑥 ∈
𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) → ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵) |
18 | | nfv 1521 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑣 𝑦 ∈ 𝐵 |
19 | | nfcsb1v 3082 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑥⦋𝑣 / 𝑥⦌𝐵 |
20 | 19 | nfcri 2306 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑥 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵 |
21 | | csbeq1a 3058 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑣 → 𝐵 = ⦋𝑣 / 𝑥⦌𝐵) |
22 | 21 | eleq2d 2240 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑣 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) |
23 | 18, 20, 22 | cbvrex 2693 |
. . . . . . . . . . . . . . 15
⊢
(∃𝑥 ∈
𝐷 𝑦 ∈ 𝐵 ↔ ∃𝑣 ∈ 𝐷 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵) |
24 | 17, 23 | sylib 121 |
. . . . . . . . . . . . . 14
⊢
((((∀𝑥 ∈
𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) → ∃𝑣 ∈ 𝐷 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵) |
25 | | simplrl 530 |
. . . . . . . . . . . . . . 15
⊢
(((((∀𝑥
∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → 𝑢 ∈ 𝐶) |
26 | | simprl 526 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((∀𝑥 ∈
𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → 𝐶 ⊆ 𝐴) |
27 | 26 | ad3antrrr 489 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((∀𝑥
∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → 𝐶 ⊆ 𝐴) |
28 | 27, 25 | sseldd 3148 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((∀𝑥
∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → 𝑢 ∈ 𝐴) |
29 | | simprr 527 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((∀𝑥 ∈
𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → 𝐷 ⊆ 𝐴) |
30 | 29 | ad3antrrr 489 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((∀𝑥
∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → 𝐷 ⊆ 𝐴) |
31 | | simprl 526 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((∀𝑥
∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → 𝑣 ∈ 𝐷) |
32 | 30, 31 | sseldd 3148 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((∀𝑥
∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → 𝑣 ∈ 𝐴) |
33 | 28, 32 | jca 304 |
. . . . . . . . . . . . . . . . 17
⊢
(((((∀𝑥
∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) |
34 | | simp-4l 536 |
. . . . . . . . . . . . . . . . 17
⊢
(((((∀𝑥
∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → ∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤)) |
35 | | simplrr 531 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((∀𝑥
∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵) |
36 | | simprr 527 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((∀𝑥
∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵) |
37 | 20, 22 | sbie 1784 |
. . . . . . . . . . . . . . . . . . 19
⊢ ([𝑣 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵) |
38 | 36, 37 | sylibr 133 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((∀𝑥
∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → [𝑣 / 𝑥]𝑦 ∈ 𝐵) |
39 | 35, 38 | jca 304 |
. . . . . . . . . . . . . . . . 17
⊢
(((((∀𝑥
∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → (𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵 ∧ [𝑣 / 𝑥]𝑦 ∈ 𝐵)) |
40 | | nfs1v 1932 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑥[𝑤 / 𝑥]𝑦 ∈ 𝐵 |
41 | 12, 40 | nfan 1558 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑥(𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) |
42 | | nfv 1521 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑥 𝑢 = 𝑤 |
43 | 41, 42 | nfim 1565 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑥((𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑢 = 𝑤) |
44 | | nfv 1521 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑤((𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵 ∧ [𝑣 / 𝑥]𝑦 ∈ 𝐵) → 𝑢 = 𝑣) |
45 | 14 | anbi1d 462 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑢 → ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) ↔ (𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵))) |
46 | | equequ1 1705 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑢 → (𝑥 = 𝑤 ↔ 𝑢 = 𝑤)) |
47 | 45, 46 | imbi12d 233 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑢 → (((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ↔ ((𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑢 = 𝑤))) |
48 | | sbequ 1833 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑤 = 𝑣 → ([𝑤 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝑣 / 𝑥]𝑦 ∈ 𝐵)) |
49 | 48 | anbi2d 461 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑤 = 𝑣 → ((𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) ↔ (𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵 ∧ [𝑣 / 𝑥]𝑦 ∈ 𝐵))) |
50 | | equequ2 1706 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑤 = 𝑣 → (𝑢 = 𝑤 ↔ 𝑢 = 𝑣)) |
51 | 49, 50 | imbi12d 233 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑤 = 𝑣 → (((𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑢 = 𝑤) ↔ ((𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵 ∧ [𝑣 / 𝑥]𝑦 ∈ 𝐵) → 𝑢 = 𝑣))) |
52 | 43, 44, 47, 51 | rspc2 2845 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) → ((𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵 ∧ [𝑣 / 𝑥]𝑦 ∈ 𝐵) → 𝑢 = 𝑣))) |
53 | 33, 34, 39, 52 | syl3c 63 |
. . . . . . . . . . . . . . . 16
⊢
(((((∀𝑥
∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → 𝑢 = 𝑣) |
54 | 53, 31 | eqeltrd 2247 |
. . . . . . . . . . . . . . 15
⊢
(((((∀𝑥
∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → 𝑢 ∈ 𝐷) |
55 | | inelcm 3475 |
. . . . . . . . . . . . . . 15
⊢ ((𝑢 ∈ 𝐶 ∧ 𝑢 ∈ 𝐷) → (𝐶 ∩ 𝐷) ≠ ∅) |
56 | 25, 54, 55 | syl2anc 409 |
. . . . . . . . . . . . . 14
⊢
(((((∀𝑥
∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → (𝐶 ∩ 𝐷) ≠ ∅) |
57 | 24, 56 | rexlimddv 2592 |
. . . . . . . . . . . . 13
⊢
((((∀𝑥 ∈
𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) → (𝐶 ∩ 𝐷) ≠ ∅) |
58 | 16, 57 | rexlimddv 2592 |
. . . . . . . . . . . 12
⊢
(((∀𝑥 ∈
𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) → (𝐶 ∩ 𝐷) ≠ ∅) |
59 | 58 | exp31 362 |
. . . . . . . . . . 11
⊢
(∀𝑥 ∈
𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) → ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴) → ((∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵) → (𝐶 ∩ 𝐷) ≠ ∅))) |
60 | 8, 59 | sylbi 120 |
. . . . . . . . . 10
⊢
(∃*𝑥 ∈
𝐴 𝑦 ∈ 𝐵 → ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴) → ((∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵) → (𝐶 ∩ 𝐷) ≠ ∅))) |
61 | 60 | impcom 124 |
. . . . . . . . 9
⊢ (((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴) ∧ ∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → ((∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵) → (𝐶 ∩ 𝐷) ≠ ∅)) |
62 | 6, 61 | syl5bi 151 |
. . . . . . . 8
⊢ (((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴) ∧ ∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → (𝑦 ∈ (∪
𝑥 ∈ 𝐶 𝐵 ∩ ∪
𝑥 ∈ 𝐷 𝐵) → (𝐶 ∩ 𝐷) ≠ ∅)) |
63 | 62 | necon2bd 2398 |
. . . . . . 7
⊢ (((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴) ∧ ∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → ((𝐶 ∩ 𝐷) = ∅ → ¬ 𝑦 ∈ (∪
𝑥 ∈ 𝐶 𝐵 ∩ ∪
𝑥 ∈ 𝐷 𝐵))) |
64 | 63 | impancom 258 |
. . . . . 6
⊢ (((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴) ∧ (𝐶 ∩ 𝐷) = ∅) → (∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ¬ 𝑦 ∈ (∪
𝑥 ∈ 𝐶 𝐵 ∩ ∪
𝑥 ∈ 𝐷 𝐵))) |
65 | 64 | 3impa 1189 |
. . . . 5
⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴 ∧ (𝐶 ∩ 𝐷) = ∅) → (∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ¬ 𝑦 ∈ (∪
𝑥 ∈ 𝐶 𝐵 ∩ ∪
𝑥 ∈ 𝐷 𝐵))) |
66 | 65 | alimdv 1872 |
. . . 4
⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴 ∧ (𝐶 ∩ 𝐷) = ∅) → (∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ∀𝑦 ¬ 𝑦 ∈ (∪
𝑥 ∈ 𝐶 𝐵 ∩ ∪
𝑥 ∈ 𝐷 𝐵))) |
67 | 1, 66 | syl5bi 151 |
. . 3
⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴 ∧ (𝐶 ∩ 𝐷) = ∅) → (Disj 𝑥 ∈ 𝐴 𝐵 → ∀𝑦 ¬ 𝑦 ∈ (∪
𝑥 ∈ 𝐶 𝐵 ∩ ∪
𝑥 ∈ 𝐷 𝐵))) |
68 | 67 | impcom 124 |
. 2
⊢
((Disj 𝑥
∈ 𝐴 𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴 ∧ (𝐶 ∩ 𝐷) = ∅)) → ∀𝑦 ¬ 𝑦 ∈ (∪
𝑥 ∈ 𝐶 𝐵 ∩ ∪
𝑥 ∈ 𝐷 𝐵)) |
69 | | eq0 3433 |
. 2
⊢
((∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪
𝑥 ∈ 𝐷 𝐵) = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ (∪
𝑥 ∈ 𝐶 𝐵 ∩ ∪
𝑥 ∈ 𝐷 𝐵)) |
70 | 68, 69 | sylibr 133 |
1
⊢
((Disj 𝑥
∈ 𝐴 𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴 ∧ (𝐶 ∩ 𝐷) = ∅)) → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪
𝑥 ∈ 𝐷 𝐵) = ∅) |