Step |
Hyp |
Ref |
Expression |
1 |
|
iundjiunlem.z |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑁 ) |
2 |
|
iundjiunlem.f |
⊢ 𝐹 = ( 𝑛 ∈ 𝑍 ↦ ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) ) |
3 |
|
iundjiunlem.j |
⊢ ( 𝜑 → 𝐽 ∈ 𝑍 ) |
4 |
|
iundjiunlem.k |
⊢ ( 𝜑 → 𝐾 ∈ 𝑍 ) |
5 |
|
iundjiunlem.lt |
⊢ ( 𝜑 → 𝐽 < 𝐾 ) |
6 |
|
incom |
⊢ ( ( 𝐹 ‘ 𝐽 ) ∩ ( 𝐹 ‘ 𝐾 ) ) = ( ( 𝐹 ‘ 𝐾 ) ∩ ( 𝐹 ‘ 𝐽 ) ) |
7 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐹 ‘ 𝐾 ) ) → 𝜑 ) |
8 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐹 ‘ 𝐾 ) ) → 𝑥 ∈ ( 𝐹 ‘ 𝐾 ) ) |
9 |
|
fveq2 |
⊢ ( 𝑛 = 𝐾 → ( 𝐸 ‘ 𝑛 ) = ( 𝐸 ‘ 𝐾 ) ) |
10 |
|
oveq2 |
⊢ ( 𝑛 = 𝐾 → ( 𝑁 ..^ 𝑛 ) = ( 𝑁 ..^ 𝐾 ) ) |
11 |
10
|
iuneq1d |
⊢ ( 𝑛 = 𝐾 → ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) = ∪ 𝑖 ∈ ( 𝑁 ..^ 𝐾 ) ( 𝐸 ‘ 𝑖 ) ) |
12 |
9 11
|
difeq12d |
⊢ ( 𝑛 = 𝐾 → ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) = ( ( 𝐸 ‘ 𝐾 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝐾 ) ( 𝐸 ‘ 𝑖 ) ) ) |
13 |
|
fvex |
⊢ ( 𝐸 ‘ 𝐾 ) ∈ V |
14 |
13
|
difexi |
⊢ ( ( 𝐸 ‘ 𝐾 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝐾 ) ( 𝐸 ‘ 𝑖 ) ) ∈ V |
15 |
12 2 14
|
fvmpt |
⊢ ( 𝐾 ∈ 𝑍 → ( 𝐹 ‘ 𝐾 ) = ( ( 𝐸 ‘ 𝐾 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝐾 ) ( 𝐸 ‘ 𝑖 ) ) ) |
16 |
4 15
|
syl |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐾 ) = ( ( 𝐸 ‘ 𝐾 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝐾 ) ( 𝐸 ‘ 𝑖 ) ) ) |
17 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐹 ‘ 𝐾 ) ) → ( 𝐹 ‘ 𝐾 ) = ( ( 𝐸 ‘ 𝐾 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝐾 ) ( 𝐸 ‘ 𝑖 ) ) ) |
18 |
8 17
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐹 ‘ 𝐾 ) ) → 𝑥 ∈ ( ( 𝐸 ‘ 𝐾 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝐾 ) ( 𝐸 ‘ 𝑖 ) ) ) |
19 |
18
|
eldifbd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐹 ‘ 𝐾 ) ) → ¬ 𝑥 ∈ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝐾 ) ( 𝐸 ‘ 𝑖 ) ) |
20 |
3 1
|
eleqtrdi |
⊢ ( 𝜑 → 𝐽 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
21 |
1 4
|
eluzelz2d |
⊢ ( 𝜑 → 𝐾 ∈ ℤ ) |
22 |
20 21 5
|
elfzod |
⊢ ( 𝜑 → 𝐽 ∈ ( 𝑁 ..^ 𝐾 ) ) |
23 |
|
fveq2 |
⊢ ( 𝑖 = 𝐽 → ( 𝐸 ‘ 𝑖 ) = ( 𝐸 ‘ 𝐽 ) ) |
24 |
23
|
ssiun2s |
⊢ ( 𝐽 ∈ ( 𝑁 ..^ 𝐾 ) → ( 𝐸 ‘ 𝐽 ) ⊆ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝐾 ) ( 𝐸 ‘ 𝑖 ) ) |
25 |
22 24
|
syl |
⊢ ( 𝜑 → ( 𝐸 ‘ 𝐽 ) ⊆ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝐾 ) ( 𝐸 ‘ 𝑖 ) ) |
26 |
25
|
ssneld |
⊢ ( 𝜑 → ( ¬ 𝑥 ∈ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝐾 ) ( 𝐸 ‘ 𝑖 ) → ¬ 𝑥 ∈ ( 𝐸 ‘ 𝐽 ) ) ) |
27 |
7 19 26
|
sylc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐹 ‘ 𝐾 ) ) → ¬ 𝑥 ∈ ( 𝐸 ‘ 𝐽 ) ) |
28 |
|
eldifi |
⊢ ( 𝑥 ∈ ( ( 𝐸 ‘ 𝐽 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝐽 ) ( 𝐸 ‘ 𝑖 ) ) → 𝑥 ∈ ( 𝐸 ‘ 𝐽 ) ) |
29 |
27 28
|
nsyl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐹 ‘ 𝐾 ) ) → ¬ 𝑥 ∈ ( ( 𝐸 ‘ 𝐽 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝐽 ) ( 𝐸 ‘ 𝑖 ) ) ) |
30 |
|
fveq2 |
⊢ ( 𝑛 = 𝐽 → ( 𝐸 ‘ 𝑛 ) = ( 𝐸 ‘ 𝐽 ) ) |
31 |
|
oveq2 |
⊢ ( 𝑛 = 𝐽 → ( 𝑁 ..^ 𝑛 ) = ( 𝑁 ..^ 𝐽 ) ) |
32 |
31
|
iuneq1d |
⊢ ( 𝑛 = 𝐽 → ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) = ∪ 𝑖 ∈ ( 𝑁 ..^ 𝐽 ) ( 𝐸 ‘ 𝑖 ) ) |
33 |
30 32
|
difeq12d |
⊢ ( 𝑛 = 𝐽 → ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) = ( ( 𝐸 ‘ 𝐽 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝐽 ) ( 𝐸 ‘ 𝑖 ) ) ) |
34 |
|
fvex |
⊢ ( 𝐸 ‘ 𝐽 ) ∈ V |
35 |
34
|
difexi |
⊢ ( ( 𝐸 ‘ 𝐽 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝐽 ) ( 𝐸 ‘ 𝑖 ) ) ∈ V |
36 |
33 2 35
|
fvmpt |
⊢ ( 𝐽 ∈ 𝑍 → ( 𝐹 ‘ 𝐽 ) = ( ( 𝐸 ‘ 𝐽 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝐽 ) ( 𝐸 ‘ 𝑖 ) ) ) |
37 |
3 36
|
syl |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐽 ) = ( ( 𝐸 ‘ 𝐽 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝐽 ) ( 𝐸 ‘ 𝑖 ) ) ) |
38 |
37
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐹 ‘ 𝐾 ) ) → ( 𝐹 ‘ 𝐽 ) = ( ( 𝐸 ‘ 𝐽 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝐽 ) ( 𝐸 ‘ 𝑖 ) ) ) |
39 |
29 38
|
neleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐹 ‘ 𝐾 ) ) → ¬ 𝑥 ∈ ( 𝐹 ‘ 𝐽 ) ) |
40 |
39
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ( 𝐹 ‘ 𝐾 ) ¬ 𝑥 ∈ ( 𝐹 ‘ 𝐽 ) ) |
41 |
|
disj |
⊢ ( ( ( 𝐹 ‘ 𝐾 ) ∩ ( 𝐹 ‘ 𝐽 ) ) = ∅ ↔ ∀ 𝑥 ∈ ( 𝐹 ‘ 𝐾 ) ¬ 𝑥 ∈ ( 𝐹 ‘ 𝐽 ) ) |
42 |
40 41
|
sylibr |
⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝐾 ) ∩ ( 𝐹 ‘ 𝐽 ) ) = ∅ ) |
43 |
6 42
|
syl5eq |
⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝐽 ) ∩ ( 𝐹 ‘ 𝐾 ) ) = ∅ ) |