Step |
Hyp |
Ref |
Expression |
1 |
|
inf3lem.1 |
⊢ 𝐺 = ( 𝑦 ∈ V ↦ { 𝑤 ∈ 𝑥 ∣ ( 𝑤 ∩ 𝑥 ) ⊆ 𝑦 } ) |
2 |
|
inf3lem.2 |
⊢ 𝐹 = ( rec ( 𝐺 , ∅ ) ↾ ω ) |
3 |
|
inf3lem.3 |
⊢ 𝐴 ∈ V |
4 |
|
inf3lem.4 |
⊢ 𝐵 ∈ V |
5 |
1 2 3 4
|
inf3lem1 |
⊢ ( 𝐴 ∈ ω → ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ suc 𝐴 ) ) |
6 |
5
|
a1i |
⊢ ( ( 𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥 ) → ( 𝐴 ∈ ω → ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ suc 𝐴 ) ) ) |
7 |
1 2 3 4
|
inf3lem3 |
⊢ ( ( 𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥 ) → ( 𝐴 ∈ ω → ( 𝐹 ‘ 𝐴 ) ≠ ( 𝐹 ‘ suc 𝐴 ) ) ) |
8 |
6 7
|
jcad |
⊢ ( ( 𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥 ) → ( 𝐴 ∈ ω → ( ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ suc 𝐴 ) ∧ ( 𝐹 ‘ 𝐴 ) ≠ ( 𝐹 ‘ suc 𝐴 ) ) ) ) |
9 |
|
df-pss |
⊢ ( ( 𝐹 ‘ 𝐴 ) ⊊ ( 𝐹 ‘ suc 𝐴 ) ↔ ( ( 𝐹 ‘ 𝐴 ) ⊆ ( 𝐹 ‘ suc 𝐴 ) ∧ ( 𝐹 ‘ 𝐴 ) ≠ ( 𝐹 ‘ suc 𝐴 ) ) ) |
10 |
8 9
|
syl6ibr |
⊢ ( ( 𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥 ) → ( 𝐴 ∈ ω → ( 𝐹 ‘ 𝐴 ) ⊊ ( 𝐹 ‘ suc 𝐴 ) ) ) |