Proof of Theorem unblem1
| Step | Hyp | Ref
| Expression |
| 1 | | omsson 7891 |
. . . . . 6
⊢ ω
⊆ On |
| 2 | | sstr 3992 |
. . . . . 6
⊢ ((𝐵 ⊆ ω ∧ ω
⊆ On) → 𝐵
⊆ On) |
| 3 | 1, 2 | mpan2 691 |
. . . . 5
⊢ (𝐵 ⊆ ω → 𝐵 ⊆ On) |
| 4 | 3 | ssdifssd 4147 |
. . . 4
⊢ (𝐵 ⊆ ω → (𝐵 ∖ suc 𝐴) ⊆ On) |
| 5 | 4 | ad2antrr 726 |
. . 3
⊢ (((𝐵 ⊆ ω ∧
∀𝑥 ∈ ω
∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) ∧ 𝐴 ∈ 𝐵) → (𝐵 ∖ suc 𝐴) ⊆ On) |
| 6 | | ssel 3977 |
. . . . . 6
⊢ (𝐵 ⊆ ω → (𝐴 ∈ 𝐵 → 𝐴 ∈ ω)) |
| 7 | | peano2b 7904 |
. . . . . 6
⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈
ω) |
| 8 | 6, 7 | imbitrdi 251 |
. . . . 5
⊢ (𝐵 ⊆ ω → (𝐴 ∈ 𝐵 → suc 𝐴 ∈ ω)) |
| 9 | | eleq1 2829 |
. . . . . . . 8
⊢ (𝑥 = suc 𝐴 → (𝑥 ∈ 𝑦 ↔ suc 𝐴 ∈ 𝑦)) |
| 10 | 9 | rexbidv 3179 |
. . . . . . 7
⊢ (𝑥 = suc 𝐴 → (∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 ↔ ∃𝑦 ∈ 𝐵 suc 𝐴 ∈ 𝑦)) |
| 11 | 10 | rspccva 3621 |
. . . . . 6
⊢
((∀𝑥 ∈
ω ∃𝑦 ∈
𝐵 𝑥 ∈ 𝑦 ∧ suc 𝐴 ∈ ω) → ∃𝑦 ∈ 𝐵 suc 𝐴 ∈ 𝑦) |
| 12 | | ssel 3977 |
. . . . . . . . . . 11
⊢ (𝐵 ⊆ ω → (𝑦 ∈ 𝐵 → 𝑦 ∈ ω)) |
| 13 | | nnord 7895 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ω → Ord 𝑦) |
| 14 | | ordn2lp 6404 |
. . . . . . . . . . . . . 14
⊢ (Ord
𝑦 → ¬ (𝑦 ∈ suc 𝐴 ∧ suc 𝐴 ∈ 𝑦)) |
| 15 | | imnan 399 |
. . . . . . . . . . . . . 14
⊢ ((𝑦 ∈ suc 𝐴 → ¬ suc 𝐴 ∈ 𝑦) ↔ ¬ (𝑦 ∈ suc 𝐴 ∧ suc 𝐴 ∈ 𝑦)) |
| 16 | 14, 15 | sylibr 234 |
. . . . . . . . . . . . 13
⊢ (Ord
𝑦 → (𝑦 ∈ suc 𝐴 → ¬ suc 𝐴 ∈ 𝑦)) |
| 17 | 16 | con2d 134 |
. . . . . . . . . . . 12
⊢ (Ord
𝑦 → (suc 𝐴 ∈ 𝑦 → ¬ 𝑦 ∈ suc 𝐴)) |
| 18 | 13, 17 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ω → (suc
𝐴 ∈ 𝑦 → ¬ 𝑦 ∈ suc 𝐴)) |
| 19 | 12, 18 | syl6 35 |
. . . . . . . . . 10
⊢ (𝐵 ⊆ ω → (𝑦 ∈ 𝐵 → (suc 𝐴 ∈ 𝑦 → ¬ 𝑦 ∈ suc 𝐴))) |
| 20 | 19 | imdistand 570 |
. . . . . . . . 9
⊢ (𝐵 ⊆ ω → ((𝑦 ∈ 𝐵 ∧ suc 𝐴 ∈ 𝑦) → (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ suc 𝐴))) |
| 21 | | eldif 3961 |
. . . . . . . . . 10
⊢ (𝑦 ∈ (𝐵 ∖ suc 𝐴) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ suc 𝐴)) |
| 22 | | ne0i 4341 |
. . . . . . . . . 10
⊢ (𝑦 ∈ (𝐵 ∖ suc 𝐴) → (𝐵 ∖ suc 𝐴) ≠ ∅) |
| 23 | 21, 22 | sylbir 235 |
. . . . . . . . 9
⊢ ((𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ suc 𝐴) → (𝐵 ∖ suc 𝐴) ≠ ∅) |
| 24 | 20, 23 | syl6 35 |
. . . . . . . 8
⊢ (𝐵 ⊆ ω → ((𝑦 ∈ 𝐵 ∧ suc 𝐴 ∈ 𝑦) → (𝐵 ∖ suc 𝐴) ≠ ∅)) |
| 25 | 24 | expd 415 |
. . . . . . 7
⊢ (𝐵 ⊆ ω → (𝑦 ∈ 𝐵 → (suc 𝐴 ∈ 𝑦 → (𝐵 ∖ suc 𝐴) ≠ ∅))) |
| 26 | 25 | rexlimdv 3153 |
. . . . . 6
⊢ (𝐵 ⊆ ω →
(∃𝑦 ∈ 𝐵 suc 𝐴 ∈ 𝑦 → (𝐵 ∖ suc 𝐴) ≠ ∅)) |
| 27 | 11, 26 | syl5 34 |
. . . . 5
⊢ (𝐵 ⊆ ω →
((∀𝑥 ∈ ω
∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 ∧ suc 𝐴 ∈ ω) → (𝐵 ∖ suc 𝐴) ≠ ∅)) |
| 28 | 8, 27 | sylan2d 605 |
. . . 4
⊢ (𝐵 ⊆ ω →
((∀𝑥 ∈ ω
∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 ∧ 𝐴 ∈ 𝐵) → (𝐵 ∖ suc 𝐴) ≠ ∅)) |
| 29 | 28 | impl 455 |
. . 3
⊢ (((𝐵 ⊆ ω ∧
∀𝑥 ∈ ω
∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) ∧ 𝐴 ∈ 𝐵) → (𝐵 ∖ suc 𝐴) ≠ ∅) |
| 30 | | onint 7810 |
. . 3
⊢ (((𝐵 ∖ suc 𝐴) ⊆ On ∧ (𝐵 ∖ suc 𝐴) ≠ ∅) → ∩ (𝐵
∖ suc 𝐴) ∈
(𝐵 ∖ suc 𝐴)) |
| 31 | 5, 29, 30 | syl2anc 584 |
. 2
⊢ (((𝐵 ⊆ ω ∧
∀𝑥 ∈ ω
∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) ∧ 𝐴 ∈ 𝐵) → ∩ (𝐵 ∖ suc 𝐴) ∈ (𝐵 ∖ suc 𝐴)) |
| 32 | 31 | eldifad 3963 |
1
⊢ (((𝐵 ⊆ ω ∧
∀𝑥 ∈ ω
∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) ∧ 𝐴 ∈ 𝐵) → ∩ (𝐵 ∖ suc 𝐴) ∈ 𝐵) |