Step | Hyp | Ref
| Expression |
1 | | elif 4534 |
. . 3
⊢ (𝑥 ∈ if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ↔ ((𝑂 ≺ 𝐵 ∧ 𝑥 ∈ {sup(𝐵, 𝐴, 𝑅)}) ∨ (¬ 𝑂 ≺ 𝐵 ∧ 𝑥 ∈ 𝐵))) |
2 | | elsni 4608 |
. . . . . 6
⊢ (𝑥 ∈ {sup(𝐵, 𝐴, 𝑅)} → 𝑥 = sup(𝐵, 𝐴, 𝑅)) |
3 | | simpr 486 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑂 ≺ 𝐵) ∧ 𝑥 = sup(𝐵, 𝐴, 𝑅)) → 𝑥 = sup(𝐵, 𝐴, 𝑅)) |
4 | | safesnsupfiss.ordered |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑅 Or 𝐴) |
5 | 4 | adantr 482 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → 𝑅 Or 𝐴) |
6 | | safesnsupfiss.finite |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐵 ∈ Fin) |
7 | 6 | adantr 482 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → 𝐵 ∈ Fin) |
8 | | simpr 486 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → 𝑂 ≺ 𝐵) |
9 | | safesnsupfiss.small |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑂 = ∅ ∨ 𝑂 = 1o)) |
10 | | 0elon 6376 |
. . . . . . . . . . . . . . 15
⊢ ∅
∈ On |
11 | | eleq1 2826 |
. . . . . . . . . . . . . . 15
⊢ (𝑂 = ∅ → (𝑂 ∈ On ↔ ∅ ∈
On)) |
12 | 10, 11 | mpbiri 258 |
. . . . . . . . . . . . . 14
⊢ (𝑂 = ∅ → 𝑂 ∈ On) |
13 | | 1on 8429 |
. . . . . . . . . . . . . . 15
⊢
1o ∈ On |
14 | | eleq1 2826 |
. . . . . . . . . . . . . . 15
⊢ (𝑂 = 1o → (𝑂 ∈ On ↔ 1o
∈ On)) |
15 | 13, 14 | mpbiri 258 |
. . . . . . . . . . . . . 14
⊢ (𝑂 = 1o → 𝑂 ∈ On) |
16 | 12, 15 | jaoi 856 |
. . . . . . . . . . . . 13
⊢ ((𝑂 = ∅ ∨ 𝑂 = 1o) → 𝑂 ∈ On) |
17 | 9, 16 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑂 ∈ On) |
18 | 17 | adantr 482 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → 𝑂 ∈ On) |
19 | 8, 18 | sdomne0d 41760 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → 𝐵 ≠ ∅) |
20 | | safesnsupfiss.subset |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
21 | 20 | adantr 482 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → 𝐵 ⊆ 𝐴) |
22 | | fisupcl 9412 |
. . . . . . . . . 10
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵) |
23 | 5, 7, 19, 21, 22 | syl13anc 1373 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵) |
24 | 23 | adantr 482 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑂 ≺ 𝐵) ∧ 𝑥 = sup(𝐵, 𝐴, 𝑅)) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵) |
25 | 3, 24 | eqeltrd 2838 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑂 ≺ 𝐵) ∧ 𝑥 = sup(𝐵, 𝐴, 𝑅)) → 𝑥 ∈ 𝐵) |
26 | 25 | ex 414 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → (𝑥 = sup(𝐵, 𝐴, 𝑅) → 𝑥 ∈ 𝐵)) |
27 | 2, 26 | syl5 34 |
. . . . 5
⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → (𝑥 ∈ {sup(𝐵, 𝐴, 𝑅)} → 𝑥 ∈ 𝐵)) |
28 | 27 | expimpd 455 |
. . . 4
⊢ (𝜑 → ((𝑂 ≺ 𝐵 ∧ 𝑥 ∈ {sup(𝐵, 𝐴, 𝑅)}) → 𝑥 ∈ 𝐵)) |
29 | | simpr 486 |
. . . . 5
⊢ ((¬
𝑂 ≺ 𝐵 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) |
30 | 29 | a1i 11 |
. . . 4
⊢ (𝜑 → ((¬ 𝑂 ≺ 𝐵 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)) |
31 | 28, 30 | jaod 858 |
. . 3
⊢ (𝜑 → (((𝑂 ≺ 𝐵 ∧ 𝑥 ∈ {sup(𝐵, 𝐴, 𝑅)}) ∨ (¬ 𝑂 ≺ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑥 ∈ 𝐵)) |
32 | 1, 31 | biimtrid 241 |
. 2
⊢ (𝜑 → (𝑥 ∈ if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) → 𝑥 ∈ 𝐵)) |
33 | 32 | ssrdv 3955 |
1
⊢ (𝜑 → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ⊆ 𝐵) |