Step |
Hyp |
Ref |
Expression |
1 |
|
sate0 |
⊢ ( 𝑈 ∈ ( Fmla ‘ ω ) → ( ∅ Sat∈ 𝑈 ) = ( ( ( ∅ Sat ∅ ) ‘ ω ) ‘ 𝑈 ) ) |
2 |
|
peano1 |
⊢ ∅ ∈ ω |
3 |
2
|
n0ii |
⊢ ¬ ω = ∅ |
4 |
3
|
intnan |
⊢ ¬ ( 𝑥 = ∅ ∧ ω = ∅ ) |
5 |
4
|
a1i |
⊢ ( 𝑈 ∈ ( Fmla ‘ ω ) → ¬ ( 𝑥 = ∅ ∧ ω = ∅ ) ) |
6 |
|
f00 |
⊢ ( 𝑥 : ω ⟶ ∅ ↔ ( 𝑥 = ∅ ∧ ω = ∅ ) ) |
7 |
5 6
|
sylnibr |
⊢ ( 𝑈 ∈ ( Fmla ‘ ω ) → ¬ 𝑥 : ω ⟶ ∅ ) |
8 |
|
0ex |
⊢ ∅ ∈ V |
9 |
8 8
|
pm3.2i |
⊢ ( ∅ ∈ V ∧ ∅ ∈ V ) |
10 |
|
satfvel |
⊢ ( ( ( ∅ ∈ V ∧ ∅ ∈ V ) ∧ 𝑈 ∈ ( Fmla ‘ ω ) ∧ 𝑥 ∈ ( ( ( ∅ Sat ∅ ) ‘ ω ) ‘ 𝑈 ) ) → 𝑥 : ω ⟶ ∅ ) |
11 |
9 10
|
mp3an1 |
⊢ ( ( 𝑈 ∈ ( Fmla ‘ ω ) ∧ 𝑥 ∈ ( ( ( ∅ Sat ∅ ) ‘ ω ) ‘ 𝑈 ) ) → 𝑥 : ω ⟶ ∅ ) |
12 |
7 11
|
mtand |
⊢ ( 𝑈 ∈ ( Fmla ‘ ω ) → ¬ 𝑥 ∈ ( ( ( ∅ Sat ∅ ) ‘ ω ) ‘ 𝑈 ) ) |
13 |
12
|
alrimiv |
⊢ ( 𝑈 ∈ ( Fmla ‘ ω ) → ∀ 𝑥 ¬ 𝑥 ∈ ( ( ( ∅ Sat ∅ ) ‘ ω ) ‘ 𝑈 ) ) |
14 |
|
eq0 |
⊢ ( ( ( ( ∅ Sat ∅ ) ‘ ω ) ‘ 𝑈 ) = ∅ ↔ ∀ 𝑥 ¬ 𝑥 ∈ ( ( ( ∅ Sat ∅ ) ‘ ω ) ‘ 𝑈 ) ) |
15 |
13 14
|
sylibr |
⊢ ( 𝑈 ∈ ( Fmla ‘ ω ) → ( ( ( ∅ Sat ∅ ) ‘ ω ) ‘ 𝑈 ) = ∅ ) |
16 |
1 15
|
eqtrd |
⊢ ( 𝑈 ∈ ( Fmla ‘ ω ) → ( ∅ Sat∈ 𝑈 ) = ∅ ) |
17 |
|
prv |
⊢ ( ( ∅ ∈ V ∧ 𝑈 ∈ ( Fmla ‘ ω ) ) → ( ∅ ⊧ 𝑈 ↔ ( ∅ Sat∈ 𝑈 ) = ( ∅ ↑m ω ) ) ) |
18 |
8 17
|
mpan |
⊢ ( 𝑈 ∈ ( Fmla ‘ ω ) → ( ∅ ⊧ 𝑈 ↔ ( ∅ Sat∈ 𝑈 ) = ( ∅ ↑m ω ) ) ) |
19 |
2
|
ne0ii |
⊢ ω ≠ ∅ |
20 |
|
map0b |
⊢ ( ω ≠ ∅ → ( ∅ ↑m ω ) = ∅ ) |
21 |
19 20
|
mp1i |
⊢ ( 𝑈 ∈ ( Fmla ‘ ω ) → ( ∅ ↑m ω ) = ∅ ) |
22 |
21
|
eqeq2d |
⊢ ( 𝑈 ∈ ( Fmla ‘ ω ) → ( ( ∅ Sat∈ 𝑈 ) = ( ∅ ↑m ω ) ↔ ( ∅ Sat∈ 𝑈 ) = ∅ ) ) |
23 |
18 22
|
bitrd |
⊢ ( 𝑈 ∈ ( Fmla ‘ ω ) → ( ∅ ⊧ 𝑈 ↔ ( ∅ Sat∈ 𝑈 ) = ∅ ) ) |
24 |
16 23
|
mpbird |
⊢ ( 𝑈 ∈ ( Fmla ‘ ω ) → ∅ ⊧ 𝑈 ) |