| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | leftssno 27919 | . . . 4
⊢ ( L
‘𝐴) ⊆  No | 
| 2 |  | fvex 6919 | . . . . 5
⊢ ( L
‘𝐴) ∈
V | 
| 3 | 2 | elpw 4604 | . . . 4
⊢ (( L
‘𝐴) ∈ 𝒫
 No  ↔ ( L ‘𝐴) ⊆  No
) | 
| 4 | 1, 3 | mpbir 231 | . . 3
⊢ ( L
‘𝐴) ∈ 𝒫
 No | 
| 5 |  | onsno 28278 | . . . . 5
⊢ (𝐴 ∈ Ons →
𝐴 ∈  No ) | 
| 6 |  | lrcut 27941 | . . . . 5
⊢ (𝐴 ∈ 
No  → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴) | 
| 7 | 5, 6 | syl 17 | . . . 4
⊢ (𝐴 ∈ Ons → ((
L ‘𝐴) |s ( R
‘𝐴)) = 𝐴) | 
| 8 |  | elons 28276 | . . . . . 6
⊢ (𝐴 ∈ Ons ↔
(𝐴 ∈  No  ∧ ( R ‘𝐴) = ∅)) | 
| 9 | 8 | simprbi 496 | . . . . 5
⊢ (𝐴 ∈ Ons → (
R ‘𝐴) =
∅) | 
| 10 | 9 | oveq2d 7447 | . . . 4
⊢ (𝐴 ∈ Ons → ((
L ‘𝐴) |s ( R
‘𝐴)) = (( L
‘𝐴) |s
∅)) | 
| 11 | 7, 10 | eqtr3d 2779 | . . 3
⊢ (𝐴 ∈ Ons →
𝐴 = (( L ‘𝐴) |s ∅)) | 
| 12 |  | oveq1 7438 | . . . 4
⊢ (𝑎 = ( L ‘𝐴) → (𝑎 |s ∅) = (( L ‘𝐴) |s ∅)) | 
| 13 | 12 | rspceeqv 3645 | . . 3
⊢ ((( L
‘𝐴) ∈ 𝒫
 No  ∧ 𝐴 = (( L ‘𝐴) |s ∅)) → ∃𝑎 ∈ 𝒫  No 𝐴 = (𝑎 |s ∅)) | 
| 14 | 4, 11, 13 | sylancr 587 | . 2
⊢ (𝐴 ∈ Ons →
∃𝑎 ∈ 𝒫
 No 𝐴 = (𝑎 |s ∅)) | 
| 15 |  | nulssgt 27843 | . . . . . 6
⊢ (𝑎 ∈ 𝒫  No  → 𝑎 <<s ∅) | 
| 16 | 15 | scutcld 27848 | . . . . 5
⊢ (𝑎 ∈ 𝒫  No  → (𝑎 |s ∅) ∈  No
) | 
| 17 |  | eqidd 2738 | . . . . . . 7
⊢ (𝑎 ∈ 𝒫  No  → (𝑎 |s ∅) = (𝑎 |s ∅)) | 
| 18 | 15, 17 | cofcutr2d 27960 | . . . . . 6
⊢ (𝑎 ∈ 𝒫  No  → ∀𝑥 ∈ ( R ‘(𝑎 |s ∅))∃𝑦 ∈ ∅ 𝑦 ≤s 𝑥) | 
| 19 |  | rex0 4360 | . . . . . . . . . 10
⊢  ¬
∃𝑦 ∈ ∅
𝑦 ≤s 𝑥 | 
| 20 |  | jcn 162 | . . . . . . . . . 10
⊢ (𝑥 ∈ ( R ‘(𝑎 |s ∅)) → (¬
∃𝑦 ∈ ∅
𝑦 ≤s 𝑥 → ¬ (𝑥 ∈ ( R ‘(𝑎 |s ∅)) → ∃𝑦 ∈ ∅ 𝑦 ≤s 𝑥))) | 
| 21 | 19, 20 | mpi 20 | . . . . . . . . 9
⊢ (𝑥 ∈ ( R ‘(𝑎 |s ∅)) → ¬
(𝑥 ∈ ( R ‘(𝑎 |s ∅)) →
∃𝑦 ∈ ∅
𝑦 ≤s 𝑥)) | 
| 22 | 21 | con2i 139 | . . . . . . . 8
⊢ ((𝑥 ∈ ( R ‘(𝑎 |s ∅)) →
∃𝑦 ∈ ∅
𝑦 ≤s 𝑥) → ¬ 𝑥 ∈ ( R ‘(𝑎 |s ∅))) | 
| 23 | 22 | alimi 1811 | . . . . . . 7
⊢
(∀𝑥(𝑥 ∈ ( R ‘(𝑎 |s ∅)) →
∃𝑦 ∈ ∅
𝑦 ≤s 𝑥) → ∀𝑥 ¬ 𝑥 ∈ ( R ‘(𝑎 |s ∅))) | 
| 24 |  | df-ral 3062 | . . . . . . 7
⊢
(∀𝑥 ∈ (
R ‘(𝑎 |s
∅))∃𝑦 ∈
∅ 𝑦 ≤s 𝑥 ↔ ∀𝑥(𝑥 ∈ ( R ‘(𝑎 |s ∅)) → ∃𝑦 ∈ ∅ 𝑦 ≤s 𝑥)) | 
| 25 |  | eq0 4350 | . . . . . . 7
⊢ (( R
‘(𝑎 |s ∅)) =
∅ ↔ ∀𝑥
¬ 𝑥 ∈ ( R
‘(𝑎 |s
∅))) | 
| 26 | 23, 24, 25 | 3imtr4i 292 | . . . . . 6
⊢
(∀𝑥 ∈ (
R ‘(𝑎 |s
∅))∃𝑦 ∈
∅ 𝑦 ≤s 𝑥 → ( R ‘(𝑎 |s ∅)) =
∅) | 
| 27 | 18, 26 | syl 17 | . . . . 5
⊢ (𝑎 ∈ 𝒫  No  → ( R ‘(𝑎 |s ∅)) = ∅) | 
| 28 |  | elons 28276 | . . . . 5
⊢ ((𝑎 |s ∅) ∈
Ons ↔ ((𝑎
|s ∅) ∈  No  ∧ ( R ‘(𝑎 |s ∅)) =
∅)) | 
| 29 | 16, 27, 28 | sylanbrc 583 | . . . 4
⊢ (𝑎 ∈ 𝒫  No  → (𝑎 |s ∅) ∈
Ons) | 
| 30 |  | eleq1 2829 | . . . 4
⊢ (𝐴 = (𝑎 |s ∅) → (𝐴 ∈ Ons ↔ (𝑎 |s ∅) ∈
Ons)) | 
| 31 | 29, 30 | syl5ibrcom 247 | . . 3
⊢ (𝑎 ∈ 𝒫  No  → (𝐴 = (𝑎 |s ∅) → 𝐴 ∈ Ons)) | 
| 32 | 31 | rexlimiv 3148 | . 2
⊢
(∃𝑎 ∈
𝒫  No 𝐴 = (𝑎 |s ∅) → 𝐴 ∈ Ons) | 
| 33 | 14, 32 | impbii 209 | 1
⊢ (𝐴 ∈ Ons ↔
∃𝑎 ∈ 𝒫
 No 𝐴 = (𝑎 |s ∅)) |