Step |
Hyp |
Ref |
Expression |
1 |
|
satff |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑥 ∈ ω ) → ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) : ( Fmla ‘ 𝑥 ) ⟶ 𝒫 ( 𝑀 ↑m ω ) ) |
2 |
1
|
3expa |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ 𝑥 ∈ ω ) → ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) : ( Fmla ‘ 𝑥 ) ⟶ 𝒫 ( 𝑀 ↑m ω ) ) |
3 |
|
entric |
⊢ ( ( 𝑥 ∈ ω ∧ 𝑦 ∈ ω ) → ( 𝑥 ≺ 𝑦 ∨ 𝑥 ≈ 𝑦 ∨ 𝑦 ≺ 𝑥 ) ) |
4 |
3
|
adantl |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 ∈ ω ) ) → ( 𝑥 ≺ 𝑦 ∨ 𝑥 ≈ 𝑦 ∨ 𝑦 ≺ 𝑥 ) ) |
5 |
|
nnsdomo |
⊢ ( ( 𝑥 ∈ ω ∧ 𝑦 ∈ ω ) → ( 𝑥 ≺ 𝑦 ↔ 𝑥 ⊊ 𝑦 ) ) |
6 |
5
|
adantl |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 ∈ ω ) ) → ( 𝑥 ≺ 𝑦 ↔ 𝑥 ⊊ 𝑦 ) ) |
7 |
|
pm3.22 |
⊢ ( ( 𝑥 ∈ ω ∧ 𝑦 ∈ ω ) → ( 𝑦 ∈ ω ∧ 𝑥 ∈ ω ) ) |
8 |
7
|
anim2i |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 ∈ ω ) ) → ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( 𝑦 ∈ ω ∧ 𝑥 ∈ ω ) ) ) |
9 |
|
pssss |
⊢ ( 𝑥 ⊊ 𝑦 → 𝑥 ⊆ 𝑦 ) |
10 |
|
eqid |
⊢ ( 𝑀 Sat 𝐸 ) = ( 𝑀 Sat 𝐸 ) |
11 |
10
|
satfsschain |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( 𝑦 ∈ ω ∧ 𝑥 ∈ ω ) ) → ( 𝑥 ⊆ 𝑦 → ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ) ) |
12 |
11
|
imp |
⊢ ( ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( 𝑦 ∈ ω ∧ 𝑥 ∈ ω ) ) ∧ 𝑥 ⊆ 𝑦 ) → ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ) |
13 |
8 9 12
|
syl2an |
⊢ ( ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 ∈ ω ) ) ∧ 𝑥 ⊊ 𝑦 ) → ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ) |
14 |
13
|
orcd |
⊢ ( ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 ∈ ω ) ) ∧ 𝑥 ⊊ 𝑦 ) → ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ∨ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ) ) |
15 |
14
|
ex |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 ∈ ω ) ) → ( 𝑥 ⊊ 𝑦 → ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ∨ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ) ) ) |
16 |
6 15
|
sylbid |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 ∈ ω ) ) → ( 𝑥 ≺ 𝑦 → ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ∨ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ) ) ) |
17 |
|
nneneq |
⊢ ( ( 𝑥 ∈ ω ∧ 𝑦 ∈ ω ) → ( 𝑥 ≈ 𝑦 ↔ 𝑥 = 𝑦 ) ) |
18 |
17
|
adantl |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 ∈ ω ) ) → ( 𝑥 ≈ 𝑦 ↔ 𝑥 = 𝑦 ) ) |
19 |
|
ssid |
⊢ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) |
20 |
|
fveq2 |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) = ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ) |
21 |
19 20
|
sseqtrrid |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ) |
22 |
21
|
olcd |
⊢ ( 𝑥 = 𝑦 → ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ∨ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ) ) |
23 |
18 22
|
syl6bi |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 ∈ ω ) ) → ( 𝑥 ≈ 𝑦 → ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ∨ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ) ) ) |
24 |
|
nnsdomo |
⊢ ( ( 𝑦 ∈ ω ∧ 𝑥 ∈ ω ) → ( 𝑦 ≺ 𝑥 ↔ 𝑦 ⊊ 𝑥 ) ) |
25 |
24
|
ancoms |
⊢ ( ( 𝑥 ∈ ω ∧ 𝑦 ∈ ω ) → ( 𝑦 ≺ 𝑥 ↔ 𝑦 ⊊ 𝑥 ) ) |
26 |
25
|
adantl |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 ∈ ω ) ) → ( 𝑦 ≺ 𝑥 ↔ 𝑦 ⊊ 𝑥 ) ) |
27 |
10
|
satfsschain |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 ∈ ω ) ) → ( 𝑦 ⊆ 𝑥 → ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ) ) |
28 |
|
pssss |
⊢ ( 𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝑥 ) |
29 |
27 28
|
impel |
⊢ ( ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 ∈ ω ) ) ∧ 𝑦 ⊊ 𝑥 ) → ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ) |
30 |
29
|
olcd |
⊢ ( ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 ∈ ω ) ) ∧ 𝑦 ⊊ 𝑥 ) → ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ∨ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ) ) |
31 |
30
|
ex |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 ∈ ω ) ) → ( 𝑦 ⊊ 𝑥 → ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ∨ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ) ) ) |
32 |
26 31
|
sylbid |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 ∈ ω ) ) → ( 𝑦 ≺ 𝑥 → ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ∨ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ) ) ) |
33 |
16 23 32
|
3jaod |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 ∈ ω ) ) → ( ( 𝑥 ≺ 𝑦 ∨ 𝑥 ≈ 𝑦 ∨ 𝑦 ≺ 𝑥 ) → ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ∨ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ) ) ) |
34 |
4 33
|
mpd |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 ∈ ω ) ) → ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ∨ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ) ) |
35 |
34
|
expr |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ 𝑥 ∈ ω ) → ( 𝑦 ∈ ω → ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ∨ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ) ) ) |
36 |
35
|
ralrimiv |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ 𝑥 ∈ ω ) → ∀ 𝑦 ∈ ω ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ∨ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ) ) |
37 |
2 36
|
jca |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ 𝑥 ∈ ω ) → ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) : ( Fmla ‘ 𝑥 ) ⟶ 𝒫 ( 𝑀 ↑m ω ) ∧ ∀ 𝑦 ∈ ω ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ∨ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ) ) ) |
38 |
37
|
ralrimiva |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ∀ 𝑥 ∈ ω ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) : ( Fmla ‘ 𝑥 ) ⟶ 𝒫 ( 𝑀 ↑m ω ) ∧ ∀ 𝑦 ∈ ω ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ∨ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ) ) ) |
39 |
|
fvex |
⊢ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ∈ V |
40 |
20 39
|
fiun |
⊢ ( ∀ 𝑥 ∈ ω ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) : ( Fmla ‘ 𝑥 ) ⟶ 𝒫 ( 𝑀 ↑m ω ) ∧ ∀ 𝑦 ∈ ω ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ∨ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑦 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ) ) → ∪ 𝑥 ∈ ω ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) : ∪ 𝑥 ∈ ω ( Fmla ‘ 𝑥 ) ⟶ 𝒫 ( 𝑀 ↑m ω ) ) |
41 |
38 40
|
syl |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ∪ 𝑥 ∈ ω ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) : ∪ 𝑥 ∈ ω ( Fmla ‘ 𝑥 ) ⟶ 𝒫 ( 𝑀 ↑m ω ) ) |
42 |
|
satom |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( ( 𝑀 Sat 𝐸 ) ‘ ω ) = ∪ 𝑥 ∈ ω ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) ) |
43 |
|
fmla |
⊢ ( Fmla ‘ ω ) = ∪ 𝑥 ∈ ω ( Fmla ‘ 𝑥 ) |
44 |
43
|
a1i |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( Fmla ‘ ω ) = ∪ 𝑥 ∈ ω ( Fmla ‘ 𝑥 ) ) |
45 |
42 44
|
feq12d |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( ( ( 𝑀 Sat 𝐸 ) ‘ ω ) : ( Fmla ‘ ω ) ⟶ 𝒫 ( 𝑀 ↑m ω ) ↔ ∪ 𝑥 ∈ ω ( ( 𝑀 Sat 𝐸 ) ‘ 𝑥 ) : ∪ 𝑥 ∈ ω ( Fmla ‘ 𝑥 ) ⟶ 𝒫 ( 𝑀 ↑m ω ) ) ) |
46 |
41 45
|
mpbird |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( ( 𝑀 Sat 𝐸 ) ‘ ω ) : ( Fmla ‘ ω ) ⟶ 𝒫 ( 𝑀 ↑m ω ) ) |