Step |
Hyp |
Ref |
Expression |
1 |
|
suceq |
⊢ ( 𝐴 = if ( 𝐴 ∈ On , 𝐴 , ∅ ) → suc 𝐴 = suc if ( 𝐴 ∈ On , 𝐴 , ∅ ) ) |
2 |
|
suceq |
⊢ ( suc 𝐴 = suc if ( 𝐴 ∈ On , 𝐴 , ∅ ) → suc suc 𝐴 = suc suc if ( 𝐴 ∈ On , 𝐴 , ∅ ) ) |
3 |
1 2
|
syl |
⊢ ( 𝐴 = if ( 𝐴 ∈ On , 𝐴 , ∅ ) → suc suc 𝐴 = suc suc if ( 𝐴 ∈ On , 𝐴 , ∅ ) ) |
4 |
3
|
eleq1d |
⊢ ( 𝐴 = if ( 𝐴 ∈ On , 𝐴 , ∅ ) → ( suc suc 𝐴 ∈ Comp ↔ suc suc if ( 𝐴 ∈ On , 𝐴 , ∅ ) ∈ Comp ) ) |
5 |
|
0elon |
⊢ ∅ ∈ On |
6 |
5
|
elimel |
⊢ if ( 𝐴 ∈ On , 𝐴 , ∅ ) ∈ On |
7 |
6
|
onsucsuccmpi |
⊢ suc suc if ( 𝐴 ∈ On , 𝐴 , ∅ ) ∈ Comp |
8 |
4 7
|
dedth |
⊢ ( 𝐴 ∈ On → suc suc 𝐴 ∈ Comp ) |