| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | simp3l 1201 | . . 3
⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → 𝐶 <<s {(𝐴 |s 𝐵)}) | 
| 2 |  | simp3r 1202 | . . 3
⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → {(𝐴 |s 𝐵)} <<s 𝐷) | 
| 3 |  | simp1 1136 | . . . . . 6
⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → 𝐴 <<s 𝐵) | 
| 4 |  | scutbday 27850 | . . . . . 6
⊢ (𝐴 <<s 𝐵 → ( bday
‘(𝐴 |s 𝐵)) = ∩ ( bday  “ {𝑡 ∈ 
No  ∣ (𝐴
<<s {𝑡} ∧ {𝑡} <<s 𝐵)})) | 
| 5 | 3, 4 | syl 17 | . . . . 5
⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → ( bday
‘(𝐴 |s 𝐵)) = ∩ ( bday  “ {𝑡 ∈ 
No  ∣ (𝐴
<<s {𝑡} ∧ {𝑡} <<s 𝐵)})) | 
| 6 |  | ssltex1 27832 | . . . . . . . . . . . . 13
⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V) | 
| 7 | 3, 6 | syl 17 | . . . . . . . . . . . 12
⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → 𝐴 ∈ V) | 
| 8 | 7 | ad2antrr 726 | . . . . . . . . . . 11
⊢ ((((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈  No )
∧ 𝐶 <<s {𝑡}) → 𝐴 ∈ V) | 
| 9 |  | ssltss1 27834 | . . . . . . . . . . . . 13
⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆  No
) | 
| 10 | 3, 9 | syl 17 | . . . . . . . . . . . 12
⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → 𝐴 ⊆  No
) | 
| 11 | 10 | ad2antrr 726 | . . . . . . . . . . 11
⊢ ((((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈  No )
∧ 𝐶 <<s {𝑡}) → 𝐴 ⊆  No
) | 
| 12 | 8, 11 | elpwd 4605 | . . . . . . . . . 10
⊢ ((((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈  No )
∧ 𝐶 <<s {𝑡}) → 𝐴 ∈ 𝒫  No
) | 
| 13 |  | simpl2l 1226 | . . . . . . . . . . 11
⊢ (((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈  No )
→ ∀𝑥 ∈
𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦) | 
| 14 | 13 | adantr 480 | . . . . . . . . . 10
⊢ ((((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈  No )
∧ 𝐶 <<s {𝑡}) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦) | 
| 15 |  | simpr 484 | . . . . . . . . . 10
⊢ ((((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈  No )
∧ 𝐶 <<s {𝑡}) → 𝐶 <<s {𝑡}) | 
| 16 |  | cofsslt 27953 | . . . . . . . . . 10
⊢ ((𝐴 ∈ 𝒫  No  ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ 𝐶 <<s {𝑡}) → 𝐴 <<s {𝑡}) | 
| 17 | 12, 14, 15, 16 | syl3anc 1372 | . . . . . . . . 9
⊢ ((((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈  No )
∧ 𝐶 <<s {𝑡}) → 𝐴 <<s {𝑡}) | 
| 18 | 17 | ex 412 | . . . . . . . 8
⊢ (((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈  No )
→ (𝐶 <<s {𝑡} → 𝐴 <<s {𝑡})) | 
| 19 |  | ssltex2 27833 | . . . . . . . . . . . . 13
⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V) | 
| 20 | 3, 19 | syl 17 | . . . . . . . . . . . 12
⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → 𝐵 ∈ V) | 
| 21 | 20 | ad2antrr 726 | . . . . . . . . . . 11
⊢ ((((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈  No )
∧ {𝑡} <<s 𝐷) → 𝐵 ∈ V) | 
| 22 |  | ssltss2 27835 | . . . . . . . . . . . . 13
⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆  No
) | 
| 23 | 3, 22 | syl 17 | . . . . . . . . . . . 12
⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → 𝐵 ⊆  No
) | 
| 24 | 23 | ad2antrr 726 | . . . . . . . . . . 11
⊢ ((((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈  No )
∧ {𝑡} <<s 𝐷) → 𝐵 ⊆  No
) | 
| 25 | 21, 24 | elpwd 4605 | . . . . . . . . . 10
⊢ ((((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈  No )
∧ {𝑡} <<s 𝐷) → 𝐵 ∈ 𝒫  No
) | 
| 26 |  | simpl2r 1227 | . . . . . . . . . . 11
⊢ (((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈  No )
→ ∀𝑧 ∈
𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) | 
| 27 | 26 | adantr 480 | . . . . . . . . . 10
⊢ ((((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈  No )
∧ {𝑡} <<s 𝐷) → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) | 
| 28 |  | simpr 484 | . . . . . . . . . 10
⊢ ((((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈  No )
∧ {𝑡} <<s 𝐷) → {𝑡} <<s 𝐷) | 
| 29 |  | coinitsslt 27954 | . . . . . . . . . 10
⊢ ((𝐵 ∈ 𝒫  No  ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧 ∧ {𝑡} <<s 𝐷) → {𝑡} <<s 𝐵) | 
| 30 | 25, 27, 28, 29 | syl3anc 1372 | . . . . . . . . 9
⊢ ((((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈  No )
∧ {𝑡} <<s 𝐷) → {𝑡} <<s 𝐵) | 
| 31 | 30 | ex 412 | . . . . . . . 8
⊢ (((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈  No )
→ ({𝑡} <<s 𝐷 → {𝑡} <<s 𝐵)) | 
| 32 | 18, 31 | anim12d 609 | . . . . . . 7
⊢ (((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) ∧ 𝑡 ∈  No )
→ ((𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷) → (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵))) | 
| 33 | 32 | ss2rabdv 4075 | . . . . . 6
⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → {𝑡 ∈  No 
∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)} ⊆ {𝑡 ∈  No 
∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}) | 
| 34 |  | imass2 6119 | . . . . . 6
⊢ ({𝑡 ∈ 
No  ∣ (𝐶
<<s {𝑡} ∧ {𝑡} <<s 𝐷)} ⊆ {𝑡 ∈  No 
∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)} → ( bday
 “ {𝑡 ∈
 No  ∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)}) ⊆ ( bday
 “ {𝑡 ∈
 No  ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)})) | 
| 35 |  | intss 4968 | . . . . . 6
⊢ (( bday  “ {𝑡 ∈  No 
∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)}) ⊆ ( bday
 “ {𝑡 ∈
 No  ∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}) → ∩
( bday  “ {𝑡 ∈  No 
∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}) ⊆ ∩
( bday  “ {𝑡 ∈  No 
∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)})) | 
| 36 | 33, 34, 35 | 3syl 18 | . . . . 5
⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → ∩
( bday  “ {𝑡 ∈  No 
∣ (𝐴 <<s {𝑡} ∧ {𝑡} <<s 𝐵)}) ⊆ ∩
( bday  “ {𝑡 ∈  No 
∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)})) | 
| 37 | 5, 36 | eqsstrd 4017 | . . . 4
⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → ( bday
‘(𝐴 |s 𝐵)) ⊆ ∩ ( bday  “ {𝑡 ∈ 
No  ∣ (𝐶
<<s {𝑡} ∧ {𝑡} <<s 𝐷)})) | 
| 38 |  | bdayfn 27819 | . . . . . 6
⊢  bday  Fn  No | 
| 39 |  | ssrab2 4079 | . . . . . 6
⊢ {𝑡 ∈ 
No  ∣ (𝐶
<<s {𝑡} ∧ {𝑡} <<s 𝐷)} ⊆  No | 
| 40 |  | sneq 4635 | . . . . . . . . 9
⊢ (𝑡 = (𝐴 |s 𝐵) → {𝑡} = {(𝐴 |s 𝐵)}) | 
| 41 | 40 | breq2d 5154 | . . . . . . . 8
⊢ (𝑡 = (𝐴 |s 𝐵) → (𝐶 <<s {𝑡} ↔ 𝐶 <<s {(𝐴 |s 𝐵)})) | 
| 42 | 40 | breq1d 5152 | . . . . . . . 8
⊢ (𝑡 = (𝐴 |s 𝐵) → ({𝑡} <<s 𝐷 ↔ {(𝐴 |s 𝐵)} <<s 𝐷)) | 
| 43 | 41, 42 | anbi12d 632 | . . . . . . 7
⊢ (𝑡 = (𝐴 |s 𝐵) → ((𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷) ↔ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷))) | 
| 44 | 3 | scutcld 27849 | . . . . . . 7
⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → (𝐴 |s 𝐵) ∈  No
) | 
| 45 |  | simp3 1138 | . . . . . . 7
⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) | 
| 46 | 43, 44, 45 | elrabd 3693 | . . . . . 6
⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → (𝐴 |s 𝐵) ∈ {𝑡 ∈  No 
∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)}) | 
| 47 |  | fnfvima 7254 | . . . . . 6
⊢ (( bday  Fn  No  ∧ {𝑡 ∈ 
No  ∣ (𝐶
<<s {𝑡} ∧ {𝑡} <<s 𝐷)} ⊆  No 
∧ (𝐴 |s 𝐵) ∈ {𝑡 ∈  No 
∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)}) → ( bday
‘(𝐴 |s 𝐵)) ∈ ( bday  “ {𝑡 ∈  No 
∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)})) | 
| 48 | 38, 39, 46, 47 | mp3an12i 1466 | . . . . 5
⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → ( bday
‘(𝐴 |s 𝐵)) ∈ ( bday  “ {𝑡 ∈  No 
∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)})) | 
| 49 |  | intss1 4962 | . . . . 5
⊢ (( bday ‘(𝐴 |s 𝐵)) ∈ ( bday
 “ {𝑡 ∈
 No  ∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)}) → ∩
( bday  “ {𝑡 ∈  No 
∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)}) ⊆ ( bday
‘(𝐴 |s 𝐵))) | 
| 50 | 48, 49 | syl 17 | . . . 4
⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → ∩
( bday  “ {𝑡 ∈  No 
∣ (𝐶 <<s {𝑡} ∧ {𝑡} <<s 𝐷)}) ⊆ ( bday
‘(𝐴 |s 𝐵))) | 
| 51 | 37, 50 | eqssd 4000 | . . 3
⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → ( bday
‘(𝐴 |s 𝐵)) = ∩ ( bday  “ {𝑡 ∈ 
No  ∣ (𝐶
<<s {𝑡} ∧ {𝑡} <<s 𝐷)})) | 
| 52 |  | ovex 7465 | . . . . . . 7
⊢ (𝐴 |s 𝐵) ∈ V | 
| 53 | 52 | snnz 4775 | . . . . . 6
⊢ {(𝐴 |s 𝐵)} ≠ ∅ | 
| 54 |  | sslttr 27853 | . . . . . 6
⊢ ((𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷 ∧ {(𝐴 |s 𝐵)} ≠ ∅) → 𝐶 <<s 𝐷) | 
| 55 | 53, 54 | mp3an3 1451 | . . . . 5
⊢ ((𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → 𝐶 <<s 𝐷) | 
| 56 | 55 | 3ad2ant3 1135 | . . . 4
⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → 𝐶 <<s 𝐷) | 
| 57 |  | eqscut 27851 | . . . 4
⊢ ((𝐶 <<s 𝐷 ∧ (𝐴 |s 𝐵) ∈  No )
→ ((𝐶 |s 𝐷) = (𝐴 |s 𝐵) ↔ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷 ∧ ( bday
‘(𝐴 |s 𝐵)) = ∩ ( bday  “ {𝑡 ∈ 
No  ∣ (𝐶
<<s {𝑡} ∧ {𝑡} <<s 𝐷)})))) | 
| 58 | 56, 44, 57 | syl2anc 584 | . . 3
⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → ((𝐶 |s 𝐷) = (𝐴 |s 𝐵) ↔ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷 ∧ ( bday
‘(𝐴 |s 𝐵)) = ∩ ( bday  “ {𝑡 ∈ 
No  ∣ (𝐶
<<s {𝑡} ∧ {𝑡} <<s 𝐷)})))) | 
| 59 | 1, 2, 51, 58 | mpbir3and 1342 | . 2
⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → (𝐶 |s 𝐷) = (𝐴 |s 𝐵)) | 
| 60 | 59 | eqcomd 2742 | 1
⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → (𝐴 |s 𝐵) = (𝐶 |s 𝐷)) |