Step |
Hyp |
Ref |
Expression |
1 |
|
ttukeylem.1 |
⊢ ( 𝜑 → 𝐹 : ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) –1-1-onto→ ( ∪ 𝐴 ∖ 𝐵 ) ) |
2 |
|
ttukeylem.2 |
⊢ ( 𝜑 → 𝐵 ∈ 𝐴 ) |
3 |
|
ttukeylem.3 |
⊢ ( 𝜑 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ ( 𝒫 𝑥 ∩ Fin ) ⊆ 𝐴 ) ) |
4 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐷 ⊆ 𝐶 ) → 𝐷 ⊆ 𝐶 ) |
5 |
4
|
sspwd |
⊢ ( ( 𝜑 ∧ 𝐷 ⊆ 𝐶 ) → 𝒫 𝐷 ⊆ 𝒫 𝐶 ) |
6 |
|
ssrin |
⊢ ( 𝒫 𝐷 ⊆ 𝒫 𝐶 → ( 𝒫 𝐷 ∩ Fin ) ⊆ ( 𝒫 𝐶 ∩ Fin ) ) |
7 |
|
sstr2 |
⊢ ( ( 𝒫 𝐷 ∩ Fin ) ⊆ ( 𝒫 𝐶 ∩ Fin ) → ( ( 𝒫 𝐶 ∩ Fin ) ⊆ 𝐴 → ( 𝒫 𝐷 ∩ Fin ) ⊆ 𝐴 ) ) |
8 |
5 6 7
|
3syl |
⊢ ( ( 𝜑 ∧ 𝐷 ⊆ 𝐶 ) → ( ( 𝒫 𝐶 ∩ Fin ) ⊆ 𝐴 → ( 𝒫 𝐷 ∩ Fin ) ⊆ 𝐴 ) ) |
9 |
1 2 3
|
ttukeylem1 |
⊢ ( 𝜑 → ( 𝐶 ∈ 𝐴 ↔ ( 𝒫 𝐶 ∩ Fin ) ⊆ 𝐴 ) ) |
10 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐷 ⊆ 𝐶 ) → ( 𝐶 ∈ 𝐴 ↔ ( 𝒫 𝐶 ∩ Fin ) ⊆ 𝐴 ) ) |
11 |
1 2 3
|
ttukeylem1 |
⊢ ( 𝜑 → ( 𝐷 ∈ 𝐴 ↔ ( 𝒫 𝐷 ∩ Fin ) ⊆ 𝐴 ) ) |
12 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐷 ⊆ 𝐶 ) → ( 𝐷 ∈ 𝐴 ↔ ( 𝒫 𝐷 ∩ Fin ) ⊆ 𝐴 ) ) |
13 |
8 10 12
|
3imtr4d |
⊢ ( ( 𝜑 ∧ 𝐷 ⊆ 𝐶 ) → ( 𝐶 ∈ 𝐴 → 𝐷 ∈ 𝐴 ) ) |
14 |
13
|
impancom |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ 𝐴 ) → ( 𝐷 ⊆ 𝐶 → 𝐷 ∈ 𝐴 ) ) |
15 |
14
|
impr |
⊢ ( ( 𝜑 ∧ ( 𝐶 ∈ 𝐴 ∧ 𝐷 ⊆ 𝐶 ) ) → 𝐷 ∈ 𝐴 ) |