| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | simp2l 1199 | . . . 4
⊢ ((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) → 𝐴 ⊆  No
) | 
| 2 |  | simp3 1138 | . . . 4
⊢ ((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) → 𝑈 ∈ 𝐴) | 
| 3 | 1, 2 | sseldd 3983 | . . 3
⊢ ((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) → 𝑈 ∈  No
) | 
| 4 |  | nosupbnd1.1 | . . . . . 6
⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {〈dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o〉}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) | 
| 5 | 4 | nosupno 27749 | . . . . 5
⊢ ((𝐴 ⊆ 
No  ∧ 𝐴 ∈
V) → 𝑆 ∈  No ) | 
| 6 | 5 | 3ad2ant2 1134 | . . . 4
⊢ ((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) → 𝑆 ∈  No
) | 
| 7 |  | nodmon 27696 | . . . 4
⊢ (𝑆 ∈ 
No  → dom 𝑆
∈ On) | 
| 8 | 6, 7 | syl 17 | . . 3
⊢ ((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) → dom 𝑆 ∈ On) | 
| 9 |  | noreson 27706 | . . 3
⊢ ((𝑈 ∈ 
No  ∧ dom 𝑆
∈ On) → (𝑈
↾ dom 𝑆) ∈  No ) | 
| 10 | 3, 8, 9 | syl2anc 584 | . 2
⊢ ((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) → (𝑈 ↾ dom 𝑆) ∈  No
) | 
| 11 |  | dmres 6029 | . . . 4
⊢ dom
(𝑈 ↾ dom 𝑆) = (dom 𝑆 ∩ dom 𝑈) | 
| 12 |  | inss1 4236 | . . . 4
⊢ (dom
𝑆 ∩ dom 𝑈) ⊆ dom 𝑆 | 
| 13 | 11, 12 | eqsstri 4029 | . . 3
⊢ dom
(𝑈 ↾ dom 𝑆) ⊆ dom 𝑆 | 
| 14 | 13 | a1i 11 | . 2
⊢ ((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) → dom (𝑈 ↾ dom 𝑆) ⊆ dom 𝑆) | 
| 15 |  | ssidd 4006 | . 2
⊢ ((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) → dom 𝑆 ⊆ dom 𝑆) | 
| 16 |  | iffalse 4533 | . . . . . . . . . . . 12
⊢ (¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {〈dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o〉}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) = (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) | 
| 17 | 4, 16 | eqtrid 2788 | . . . . . . . . . . 11
⊢ (¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → 𝑆 = (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) | 
| 18 | 17 | dmeqd 5915 | . . . . . . . . . 10
⊢ (¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = dom (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) | 
| 19 |  | iotaex 6533 | . . . . . . . . . . 11
⊢
(℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)) ∈ V | 
| 20 |  | eqid 2736 | . . . . . . . . . . 11
⊢ (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))) = (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))) | 
| 21 | 19, 20 | dmmpti 6711 | . . . . . . . . . 10
⊢ dom
(𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))) = {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} | 
| 22 | 18, 21 | eqtrdi 2792 | . . . . . . . . 9
⊢ (¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}) | 
| 23 | 22 | eleq2d 2826 | . . . . . . . 8
⊢ (¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → (ℎ ∈ dom 𝑆 ↔ ℎ ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})) | 
| 24 |  | vex 3483 | . . . . . . . . 9
⊢ ℎ ∈ V | 
| 25 |  | eleq1w 2823 | . . . . . . . . . . . 12
⊢ (𝑦 = ℎ → (𝑦 ∈ dom 𝑢 ↔ ℎ ∈ dom 𝑢)) | 
| 26 |  | suceq 6449 | . . . . . . . . . . . . . . . 16
⊢ (𝑦 = ℎ → suc 𝑦 = suc ℎ) | 
| 27 | 26 | reseq2d 5996 | . . . . . . . . . . . . . . 15
⊢ (𝑦 = ℎ → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc ℎ)) | 
| 28 | 26 | reseq2d 5996 | . . . . . . . . . . . . . . 15
⊢ (𝑦 = ℎ → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc ℎ)) | 
| 29 | 27, 28 | eqeq12d 2752 | . . . . . . . . . . . . . 14
⊢ (𝑦 = ℎ → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))) | 
| 30 | 29 | imbi2d 340 | . . . . . . . . . . . . 13
⊢ (𝑦 = ℎ → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc ℎ) = (𝑣 ↾ suc ℎ)))) | 
| 31 | 30 | ralbidv 3177 | . . . . . . . . . . . 12
⊢ (𝑦 = ℎ → (∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc ℎ) = (𝑣 ↾ suc ℎ)))) | 
| 32 | 25, 31 | anbi12d 632 | . . . . . . . . . . 11
⊢ (𝑦 = ℎ → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ (ℎ ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) | 
| 33 | 32 | rexbidv 3178 | . . . . . . . . . 10
⊢ (𝑦 = ℎ → (∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑢 ∈ 𝐴 (ℎ ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) | 
| 34 |  | dmeq 5913 | . . . . . . . . . . . . 13
⊢ (𝑢 = 𝑝 → dom 𝑢 = dom 𝑝) | 
| 35 | 34 | eleq2d 2826 | . . . . . . . . . . . 12
⊢ (𝑢 = 𝑝 → (ℎ ∈ dom 𝑢 ↔ ℎ ∈ dom 𝑝)) | 
| 36 |  | breq2 5146 | . . . . . . . . . . . . . . 15
⊢ (𝑢 = 𝑝 → (𝑣 <s 𝑢 ↔ 𝑣 <s 𝑝)) | 
| 37 | 36 | notbid 318 | . . . . . . . . . . . . . 14
⊢ (𝑢 = 𝑝 → (¬ 𝑣 <s 𝑢 ↔ ¬ 𝑣 <s 𝑝)) | 
| 38 |  | reseq1 5990 | . . . . . . . . . . . . . . 15
⊢ (𝑢 = 𝑝 → (𝑢 ↾ suc ℎ) = (𝑝 ↾ suc ℎ)) | 
| 39 | 38 | eqeq1d 2738 | . . . . . . . . . . . . . 14
⊢ (𝑢 = 𝑝 → ((𝑢 ↾ suc ℎ) = (𝑣 ↾ suc ℎ) ↔ (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))) | 
| 40 | 37, 39 | imbi12d 344 | . . . . . . . . . . . . 13
⊢ (𝑢 = 𝑝 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc ℎ) = (𝑣 ↾ suc ℎ)) ↔ (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ)))) | 
| 41 | 40 | ralbidv 3177 | . . . . . . . . . . . 12
⊢ (𝑢 = 𝑝 → (∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc ℎ) = (𝑣 ↾ suc ℎ)) ↔ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ)))) | 
| 42 | 35, 41 | anbi12d 632 | . . . . . . . . . . 11
⊢ (𝑢 = 𝑝 → ((ℎ ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))) ↔ (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) | 
| 43 | 42 | cbvrexvw 3237 | . . . . . . . . . 10
⊢
(∃𝑢 ∈
𝐴 (ℎ ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))) ↔ ∃𝑝 ∈ 𝐴 (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ)))) | 
| 44 | 33, 43 | bitrdi 287 | . . . . . . . . 9
⊢ (𝑦 = ℎ → (∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑝 ∈ 𝐴 (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) | 
| 45 | 24, 44 | elab 3678 | . . . . . . . 8
⊢ (ℎ ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↔ ∃𝑝 ∈ 𝐴 (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ)))) | 
| 46 | 23, 45 | bitrdi 287 | . . . . . . 7
⊢ (¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → (ℎ ∈ dom 𝑆 ↔ ∃𝑝 ∈ 𝐴 (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) | 
| 47 | 46 | 3ad2ant1 1133 | . . . . . 6
⊢ ((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) → (ℎ ∈ dom 𝑆 ↔ ∃𝑝 ∈ 𝐴 (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) | 
| 48 |  | simpl1 1191 | . . . . . . . . 9
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) → ¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) | 
| 49 |  | simpl2 1192 | . . . . . . . . 9
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) → (𝐴 ⊆  No 
∧ 𝐴 ∈
V)) | 
| 50 |  | simprl 770 | . . . . . . . . 9
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) → 𝑝 ∈ 𝐴) | 
| 51 |  | simprrl 780 | . . . . . . . . 9
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) → ℎ ∈ dom 𝑝) | 
| 52 |  | simprrr 781 | . . . . . . . . 9
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) → ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))) | 
| 53 | 4 | nosupres 27753 | . . . . . . . . 9
⊢ ((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
(𝑝 ∈ 𝐴 ∧ ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ)))) → (𝑆 ↾ suc ℎ) = (𝑝 ↾ suc ℎ)) | 
| 54 | 48, 49, 50, 51, 52, 53 | syl113anc 1383 | . . . . . . . 8
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) → (𝑆 ↾ suc ℎ) = (𝑝 ↾ suc ℎ)) | 
| 55 |  | simpl2l 1226 | . . . . . . . . . . . . . . 15
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) → 𝐴 ⊆  No
) | 
| 56 | 55, 50 | sseldd 3983 | . . . . . . . . . . . . . 14
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) → 𝑝 ∈  No
) | 
| 57 | 3 | adantr 480 | . . . . . . . . . . . . . 14
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) → 𝑈 ∈  No
) | 
| 58 |  | sltso 27722 | . . . . . . . . . . . . . . 15
⊢  <s Or
 No | 
| 59 |  | soasym 5624 | . . . . . . . . . . . . . . 15
⊢ (( <s
Or  No  ∧ (𝑝 ∈  No 
∧ 𝑈 ∈  No )) → (𝑝 <s 𝑈 → ¬ 𝑈 <s 𝑝)) | 
| 60 | 58, 59 | mpan 690 | . . . . . . . . . . . . . 14
⊢ ((𝑝 ∈ 
No  ∧ 𝑈 ∈
 No ) → (𝑝 <s 𝑈 → ¬ 𝑈 <s 𝑝)) | 
| 61 | 56, 57, 60 | syl2anc 584 | . . . . . . . . . . . . 13
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) → (𝑝 <s 𝑈 → ¬ 𝑈 <s 𝑝)) | 
| 62 |  | simpl3 1193 | . . . . . . . . . . . . . 14
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) → 𝑈 ∈ 𝐴) | 
| 63 |  | breq1 5145 | . . . . . . . . . . . . . . . . 17
⊢ (𝑣 = 𝑈 → (𝑣 <s 𝑝 ↔ 𝑈 <s 𝑝)) | 
| 64 | 63 | notbid 318 | . . . . . . . . . . . . . . . 16
⊢ (𝑣 = 𝑈 → (¬ 𝑣 <s 𝑝 ↔ ¬ 𝑈 <s 𝑝)) | 
| 65 |  | reseq1 5990 | . . . . . . . . . . . . . . . . 17
⊢ (𝑣 = 𝑈 → (𝑣 ↾ suc ℎ) = (𝑈 ↾ suc ℎ)) | 
| 66 | 65 | eqeq2d 2747 | . . . . . . . . . . . . . . . 16
⊢ (𝑣 = 𝑈 → ((𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ) ↔ (𝑝 ↾ suc ℎ) = (𝑈 ↾ suc ℎ))) | 
| 67 | 64, 66 | imbi12d 344 | . . . . . . . . . . . . . . 15
⊢ (𝑣 = 𝑈 → ((¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ)) ↔ (¬ 𝑈 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑈 ↾ suc ℎ)))) | 
| 68 | 67 | rspcv 3617 | . . . . . . . . . . . . . 14
⊢ (𝑈 ∈ 𝐴 → (∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ)) → (¬ 𝑈 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑈 ↾ suc ℎ)))) | 
| 69 | 62, 52, 68 | sylc 65 | . . . . . . . . . . . . 13
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) → (¬ 𝑈 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑈 ↾ suc ℎ))) | 
| 70 | 61, 69 | syld 47 | . . . . . . . . . . . 12
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) → (𝑝 <s 𝑈 → (𝑝 ↾ suc ℎ) = (𝑈 ↾ suc ℎ))) | 
| 71 | 70 | imp 406 | . . . . . . . . . . 11
⊢ ((((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) ∧ 𝑝 <s 𝑈) → (𝑝 ↾ suc ℎ) = (𝑈 ↾ suc ℎ)) | 
| 72 |  | nodmon 27696 | . . . . . . . . . . . . . . . . 17
⊢ (𝑝 ∈ 
No  → dom 𝑝
∈ On) | 
| 73 | 56, 72 | syl 17 | . . . . . . . . . . . . . . . 16
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) → dom 𝑝 ∈ On) | 
| 74 |  | onelon 6408 | . . . . . . . . . . . . . . . 16
⊢ ((dom
𝑝 ∈ On ∧ ℎ ∈ dom 𝑝) → ℎ ∈ On) | 
| 75 | 73, 51, 74 | syl2anc 584 | . . . . . . . . . . . . . . 15
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) → ℎ ∈ On) | 
| 76 |  | onsucb 7838 | . . . . . . . . . . . . . . 15
⊢ (ℎ ∈ On ↔ suc ℎ ∈ On) | 
| 77 | 75, 76 | sylib 218 | . . . . . . . . . . . . . 14
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) → suc ℎ ∈ On) | 
| 78 |  | noreson 27706 | . . . . . . . . . . . . . 14
⊢ ((𝑈 ∈ 
No  ∧ suc ℎ
∈ On) → (𝑈
↾ suc ℎ) ∈  No ) | 
| 79 | 57, 77, 78 | syl2anc 584 | . . . . . . . . . . . . 13
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) → (𝑈 ↾ suc ℎ) ∈  No
) | 
| 80 |  | sonr 5615 | . . . . . . . . . . . . . 14
⊢ (( <s
Or  No  ∧ (𝑈 ↾ suc ℎ) ∈  No )
→ ¬ (𝑈 ↾ suc
ℎ) <s (𝑈 ↾ suc ℎ)) | 
| 81 | 58, 80 | mpan 690 | . . . . . . . . . . . . 13
⊢ ((𝑈 ↾ suc ℎ) ∈  No 
→ ¬ (𝑈 ↾ suc
ℎ) <s (𝑈 ↾ suc ℎ)) | 
| 82 | 79, 81 | syl 17 | . . . . . . . . . . . 12
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) → ¬ (𝑈 ↾ suc ℎ) <s (𝑈 ↾ suc ℎ)) | 
| 83 | 82 | adantr 480 | . . . . . . . . . . 11
⊢ ((((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) ∧ 𝑝 <s 𝑈) → ¬ (𝑈 ↾ suc ℎ) <s (𝑈 ↾ suc ℎ)) | 
| 84 | 71, 83 | eqnbrtrd 5160 | . . . . . . . . . 10
⊢ ((((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) ∧ 𝑝 <s 𝑈) → ¬ (𝑝 ↾ suc ℎ) <s (𝑈 ↾ suc ℎ)) | 
| 85 | 84 | ex 412 | . . . . . . . . 9
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) → (𝑝 <s 𝑈 → ¬ (𝑝 ↾ suc ℎ) <s (𝑈 ↾ suc ℎ))) | 
| 86 |  | sltres 27708 | . . . . . . . . . . 11
⊢ ((𝑝 ∈ 
No  ∧ 𝑈 ∈
 No  ∧ suc ℎ ∈ On) → ((𝑝 ↾ suc ℎ) <s (𝑈 ↾ suc ℎ) → 𝑝 <s 𝑈)) | 
| 87 | 56, 57, 77, 86 | syl3anc 1372 | . . . . . . . . . 10
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) → ((𝑝 ↾ suc ℎ) <s (𝑈 ↾ suc ℎ) → 𝑝 <s 𝑈)) | 
| 88 | 87 | con3d 152 | . . . . . . . . 9
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) → (¬ 𝑝 <s 𝑈 → ¬ (𝑝 ↾ suc ℎ) <s (𝑈 ↾ suc ℎ))) | 
| 89 | 85, 88 | pm2.61d 179 | . . . . . . . 8
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) → ¬ (𝑝 ↾ suc ℎ) <s (𝑈 ↾ suc ℎ)) | 
| 90 | 54, 89 | eqnbrtrd 5160 | . . . . . . 7
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))))) → ¬ (𝑆 ↾ suc ℎ) <s (𝑈 ↾ suc ℎ)) | 
| 91 | 90 | rexlimdvaa 3155 | . . . . . 6
⊢ ((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) → (∃𝑝 ∈ 𝐴 (ℎ ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc ℎ) = (𝑣 ↾ suc ℎ))) → ¬ (𝑆 ↾ suc ℎ) <s (𝑈 ↾ suc ℎ))) | 
| 92 | 47, 91 | sylbid 240 | . . . . 5
⊢ ((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) → (ℎ ∈ dom 𝑆 → ¬ (𝑆 ↾ suc ℎ) <s (𝑈 ↾ suc ℎ))) | 
| 93 | 92 | imp 406 | . . . 4
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ ℎ ∈ dom 𝑆) → ¬ (𝑆 ↾ suc ℎ) <s (𝑈 ↾ suc ℎ)) | 
| 94 |  | nodmord 27699 | . . . . . . . 8
⊢ (𝑆 ∈ 
No  → Ord dom 𝑆) | 
| 95 |  | ordsucss 7839 | . . . . . . . 8
⊢ (Ord dom
𝑆 → (ℎ ∈ dom 𝑆 → suc ℎ ⊆ dom 𝑆)) | 
| 96 | 6, 94, 95 | 3syl 18 | . . . . . . 7
⊢ ((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) → (ℎ ∈ dom 𝑆 → suc ℎ ⊆ dom 𝑆)) | 
| 97 | 96 | imp 406 | . . . . . 6
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ ℎ ∈ dom 𝑆) → suc ℎ ⊆ dom 𝑆) | 
| 98 | 97 | resabs1d 6025 | . . . . 5
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ ℎ ∈ dom 𝑆) → ((𝑈 ↾ dom 𝑆) ↾ suc ℎ) = (𝑈 ↾ suc ℎ)) | 
| 99 | 98 | breq2d 5154 | . . . 4
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ ℎ ∈ dom 𝑆) → ((𝑆 ↾ suc ℎ) <s ((𝑈 ↾ dom 𝑆) ↾ suc ℎ) ↔ (𝑆 ↾ suc ℎ) <s (𝑈 ↾ suc ℎ))) | 
| 100 | 93, 99 | mtbird 325 | . . 3
⊢ (((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) ∧ ℎ ∈ dom 𝑆) → ¬ (𝑆 ↾ suc ℎ) <s ((𝑈 ↾ dom 𝑆) ↾ suc ℎ)) | 
| 101 | 100 | ralrimiva 3145 | . 2
⊢ ((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) → ∀ℎ ∈ dom 𝑆 ¬ (𝑆 ↾ suc ℎ) <s ((𝑈 ↾ dom 𝑆) ↾ suc ℎ)) | 
| 102 |  | noresle 27743 | . 2
⊢ ((((𝑈 ↾ dom 𝑆) ∈  No 
∧ 𝑆 ∈  No ) ∧ (dom (𝑈 ↾ dom 𝑆) ⊆ dom 𝑆 ∧ dom 𝑆 ⊆ dom 𝑆 ∧ ∀ℎ ∈ dom 𝑆 ¬ (𝑆 ↾ suc ℎ) <s ((𝑈 ↾ dom 𝑆) ↾ suc ℎ))) → ¬ 𝑆 <s (𝑈 ↾ dom 𝑆)) | 
| 103 | 10, 6, 14, 15, 101, 102 | syl23anc 1378 | 1
⊢ ((¬
∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆  No 
∧ 𝐴 ∈ V) ∧
𝑈 ∈ 𝐴) → ¬ 𝑆 <s (𝑈 ↾ dom 𝑆)) |