Proof of Theorem noeta2
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | id 22 | . 2
⊢ (((𝐴 ⊆ 
No  ∧ 𝐴 ∈
𝑉) ∧ (𝐵 ⊆  No 
∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → ((𝐴 ⊆  No 
∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆  No 
∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)) | 
| 2 |  | bdayfo 27723 | . . . . . . 7
⊢  bday : No –onto→On | 
| 3 |  | fofun 6820 | . . . . . . 7
⊢ ( bday : No –onto→On → Fun 
bday ) | 
| 4 | 2, 3 | ax-mp 5 | . . . . . 6
⊢ Fun  bday | 
| 5 |  | simp1r 1198 | . . . . . . 7
⊢ (((𝐴 ⊆ 
No  ∧ 𝐴 ∈
𝑉) ∧ (𝐵 ⊆  No 
∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → 𝐴 ∈ 𝑉) | 
| 6 |  | simp2r 1200 | . . . . . . 7
⊢ (((𝐴 ⊆ 
No  ∧ 𝐴 ∈
𝑉) ∧ (𝐵 ⊆  No 
∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → 𝐵 ∈ 𝑊) | 
| 7 |  | unexg 7764 | . . . . . . 7
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V) | 
| 8 | 5, 6, 7 | syl2anc 584 | . . . . . 6
⊢ (((𝐴 ⊆ 
No  ∧ 𝐴 ∈
𝑉) ∧ (𝐵 ⊆  No 
∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → (𝐴 ∪ 𝐵) ∈ V) | 
| 9 |  | funimaexg 6652 | . . . . . 6
⊢ ((Fun
 bday  ∧ (𝐴 ∪ 𝐵) ∈ V) → (
bday  “ (𝐴
∪ 𝐵)) ∈
V) | 
| 10 | 4, 8, 9 | sylancr 587 | . . . . 5
⊢ (((𝐴 ⊆ 
No  ∧ 𝐴 ∈
𝑉) ∧ (𝐵 ⊆  No 
∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → ( bday 
“ (𝐴 ∪ 𝐵)) ∈ V) | 
| 11 | 10 | uniexd 7763 | . . . 4
⊢ (((𝐴 ⊆ 
No  ∧ 𝐴 ∈
𝑉) ∧ (𝐵 ⊆  No 
∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → ∪ ( bday  “ (𝐴 ∪ 𝐵)) ∈ V) | 
| 12 |  | imassrn 6088 | . . . . . . 7
⊢ ( bday  “ (𝐴 ∪ 𝐵)) ⊆ ran  bday | 
| 13 |  | forn 6822 | . . . . . . . 8
⊢ ( bday : No –onto→On → ran 
bday  = On) | 
| 14 | 2, 13 | ax-mp 5 | . . . . . . 7
⊢ ran  bday  = On | 
| 15 | 12, 14 | sseqtri 4031 | . . . . . 6
⊢ ( bday  “ (𝐴 ∪ 𝐵)) ⊆ On | 
| 16 |  | ssorduni 7800 | . . . . . 6
⊢ (( bday  “ (𝐴 ∪ 𝐵)) ⊆ On → Ord ∪ ( bday  “ (𝐴 ∪ 𝐵))) | 
| 17 | 15, 16 | ax-mp 5 | . . . . 5
⊢ Ord ∪ ( bday  “ (𝐴 ∪ 𝐵)) | 
| 18 |  | elon2 6394 | . . . . 5
⊢ (∪ ( bday  “ (𝐴 ∪ 𝐵)) ∈ On ↔ (Ord ∪ ( bday  “ (𝐴 ∪ 𝐵)) ∧ ∪ ( bday  “ (𝐴 ∪ 𝐵)) ∈ V)) | 
| 19 | 17, 18 | mpbiran 709 | . . . 4
⊢ (∪ ( bday  “ (𝐴 ∪ 𝐵)) ∈ On ↔ ∪ ( bday  “ (𝐴 ∪ 𝐵)) ∈ V) | 
| 20 | 11, 19 | sylibr 234 | . . 3
⊢ (((𝐴 ⊆ 
No  ∧ 𝐴 ∈
𝑉) ∧ (𝐵 ⊆  No 
∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → ∪ ( bday  “ (𝐴 ∪ 𝐵)) ∈ On) | 
| 21 |  | onsucb 7838 | . . 3
⊢ (∪ ( bday  “ (𝐴 ∪ 𝐵)) ∈ On ↔ suc ∪ ( bday  “ (𝐴 ∪ 𝐵)) ∈ On) | 
| 22 | 20, 21 | sylib 218 | . 2
⊢ (((𝐴 ⊆ 
No  ∧ 𝐴 ∈
𝑉) ∧ (𝐵 ⊆  No 
∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → suc ∪
( bday  “ (𝐴 ∪ 𝐵)) ∈ On) | 
| 23 |  | onsucuni 7849 | . . 3
⊢ (( bday  “ (𝐴 ∪ 𝐵)) ⊆ On → (
bday  “ (𝐴
∪ 𝐵)) ⊆ suc ∪ ( bday  “ (𝐴 ∪ 𝐵))) | 
| 24 | 15, 23 | mp1i 13 | . 2
⊢ (((𝐴 ⊆ 
No  ∧ 𝐴 ∈
𝑉) ∧ (𝐵 ⊆  No 
∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → ( bday 
“ (𝐴 ∪ 𝐵)) ⊆ suc ∪ ( bday  “ (𝐴 ∪ 𝐵))) | 
| 25 |  | noeta 27789 | . 2
⊢ ((((𝐴 ⊆ 
No  ∧ 𝐴 ∈
𝑉) ∧ (𝐵 ⊆  No 
∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) ∧ (suc ∪
( bday  “ (𝐴 ∪ 𝐵)) ∈ On ∧ (
bday  “ (𝐴
∪ 𝐵)) ⊆ suc ∪ ( bday  “ (𝐴 ∪ 𝐵)))) → ∃𝑧 ∈  No 
(∀𝑥 ∈ 𝐴 𝑥 <s 𝑧 ∧ ∀𝑦 ∈ 𝐵 𝑧 <s 𝑦 ∧ ( bday
‘𝑧) ⊆
suc ∪ ( bday  “
(𝐴 ∪ 𝐵)))) | 
| 26 | 1, 22, 24, 25 | syl12anc 836 | 1
⊢ (((𝐴 ⊆ 
No  ∧ 𝐴 ∈
𝑉) ∧ (𝐵 ⊆  No 
∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → ∃𝑧 ∈  No 
(∀𝑥 ∈ 𝐴 𝑥 <s 𝑧 ∧ ∀𝑦 ∈ 𝐵 𝑧 <s 𝑦 ∧ ( bday
‘𝑧) ⊆
suc ∪ ( bday  “
(𝐴 ∪ 𝐵)))) |