Proof of Theorem elicc3
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | elicc1 13432 | . 2
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | 
| 2 |  | simp1 1136 | . . . . 5
⊢ ((𝐶 ∈ ℝ*
∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐶 ∈
ℝ*) | 
| 3 | 2 | a1i 11 | . . . 4
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐶 ∈
ℝ*)) | 
| 4 |  | xrletr 13201 | . . . . . . 7
⊢ ((𝐴 ∈ ℝ*
∧ 𝐶 ∈
ℝ* ∧ 𝐵
∈ ℝ*) → ((𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐴 ≤ 𝐵)) | 
| 5 | 4 | exp5o 1355 | . . . . . 6
⊢ (𝐴 ∈ ℝ*
→ (𝐶 ∈
ℝ* → (𝐵 ∈ ℝ* → (𝐴 ≤ 𝐶 → (𝐶 ≤ 𝐵 → 𝐴 ≤ 𝐵))))) | 
| 6 | 5 | com23 86 | . . . . 5
⊢ (𝐴 ∈ ℝ*
→ (𝐵 ∈
ℝ* → (𝐶 ∈ ℝ* → (𝐴 ≤ 𝐶 → (𝐶 ≤ 𝐵 → 𝐴 ≤ 𝐵))))) | 
| 7 | 6 | imp5q 36314 | . . . 4
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐴 ≤ 𝐵)) | 
| 8 |  | df-ne 2940 | . . . . . . . . . 10
⊢ (𝐶 ≠ 𝐴 ↔ ¬ 𝐶 = 𝐴) | 
| 9 |  | xrleltne 13188 | . . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ*
∧ 𝐶 ∈
ℝ* ∧ 𝐴
≤ 𝐶) → (𝐴 < 𝐶 ↔ 𝐶 ≠ 𝐴)) | 
| 10 | 9 | biimprd 248 | . . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ*
∧ 𝐶 ∈
ℝ* ∧ 𝐴
≤ 𝐶) → (𝐶 ≠ 𝐴 → 𝐴 < 𝐶)) | 
| 11 | 8, 10 | biimtrrid 243 | . . . . . . . . 9
⊢ ((𝐴 ∈ ℝ*
∧ 𝐶 ∈
ℝ* ∧ 𝐴
≤ 𝐶) → (¬ 𝐶 = 𝐴 → 𝐴 < 𝐶)) | 
| 12 | 11 | 3adant3r3 1184 | . . . . . . . 8
⊢ ((𝐴 ∈ ℝ*
∧ (𝐶 ∈
ℝ* ∧ 𝐴
≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → (¬ 𝐶 = 𝐴 → 𝐴 < 𝐶)) | 
| 13 | 12 | adantlr 715 | . . . . . . 7
⊢ (((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → (¬ 𝐶 = 𝐴 → 𝐴 < 𝐶)) | 
| 14 |  | eqcom 2743 | . . . . . . . . . . . . . 14
⊢ (𝐶 = 𝐵 ↔ 𝐵 = 𝐶) | 
| 15 | 14 | necon3bbii 2987 | . . . . . . . . . . . . 13
⊢ (¬
𝐶 = 𝐵 ↔ 𝐵 ≠ 𝐶) | 
| 16 |  | xrleltne 13188 | . . . . . . . . . . . . . 14
⊢ ((𝐶 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐶
≤ 𝐵) → (𝐶 < 𝐵 ↔ 𝐵 ≠ 𝐶)) | 
| 17 | 16 | biimprd 248 | . . . . . . . . . . . . 13
⊢ ((𝐶 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐶
≤ 𝐵) → (𝐵 ≠ 𝐶 → 𝐶 < 𝐵)) | 
| 18 | 15, 17 | biimtrid 242 | . . . . . . . . . . . 12
⊢ ((𝐶 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐶
≤ 𝐵) → (¬ 𝐶 = 𝐵 → 𝐶 < 𝐵)) | 
| 19 | 18 | 3exp 1119 | . . . . . . . . . . 11
⊢ (𝐶 ∈ ℝ*
→ (𝐵 ∈
ℝ* → (𝐶 ≤ 𝐵 → (¬ 𝐶 = 𝐵 → 𝐶 < 𝐵)))) | 
| 20 | 19 | com12 32 | . . . . . . . . . 10
⊢ (𝐵 ∈ ℝ*
→ (𝐶 ∈
ℝ* → (𝐶 ≤ 𝐵 → (¬ 𝐶 = 𝐵 → 𝐶 < 𝐵)))) | 
| 21 | 20 | imp32 418 | . . . . . . . . 9
⊢ ((𝐵 ∈ ℝ*
∧ (𝐶 ∈
ℝ* ∧ 𝐶
≤ 𝐵)) → (¬
𝐶 = 𝐵 → 𝐶 < 𝐵)) | 
| 22 | 21 | 3adantr2 1170 | . . . . . . . 8
⊢ ((𝐵 ∈ ℝ*
∧ (𝐶 ∈
ℝ* ∧ 𝐴
≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → (¬ 𝐶 = 𝐵 → 𝐶 < 𝐵)) | 
| 23 | 22 | adantll 714 | . . . . . . 7
⊢ (((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → (¬ 𝐶 = 𝐵 → 𝐶 < 𝐵)) | 
| 24 | 13, 23 | anim12d 609 | . . . . . 6
⊢ (((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → ((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) | 
| 25 | 24 | ex 412 | . . . . 5
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → ((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))) | 
| 26 |  | df-or 848 | . . . . . 6
⊢ ((𝐶 = 𝐴 ∨ ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)) ↔ (¬ 𝐶 = 𝐴 → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵))) | 
| 27 |  | 3orass 1089 | . . . . . 6
⊢ ((𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵) ↔ (𝐶 = 𝐴 ∨ ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵))) | 
| 28 |  | pm5.6 1003 | . . . . . . 7
⊢ (((¬
𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) ↔ (¬ 𝐶 = 𝐴 → (𝐶 = 𝐵 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))) | 
| 29 |  | orcom 870 | . . . . . . . 8
⊢ ((𝐶 = 𝐵 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) ↔ ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)) | 
| 30 | 29 | imbi2i 336 | . . . . . . 7
⊢ ((¬
𝐶 = 𝐴 → (𝐶 = 𝐵 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) ↔ (¬ 𝐶 = 𝐴 → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵))) | 
| 31 | 28, 30 | bitri 275 | . . . . . 6
⊢ (((¬
𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) ↔ (¬ 𝐶 = 𝐴 → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵))) | 
| 32 | 26, 27, 31 | 3bitr4ri 304 | . . . . 5
⊢ (((¬
𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) ↔ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)) | 
| 33 | 25, 32 | imbitrdi 251 | . . . 4
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵))) | 
| 34 | 3, 7, 33 | 3jcad 1129 | . . 3
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)))) | 
| 35 |  | simp1 1136 | . . . . 5
⊢ ((𝐶 ∈ ℝ*
∧ 𝐴 ≤ 𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)) → 𝐶 ∈
ℝ*) | 
| 36 | 35 | a1i 11 | . . . 4
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)) → 𝐶 ∈
ℝ*)) | 
| 37 |  | xrleid 13194 | . . . . . . . . 9
⊢ (𝐴 ∈ ℝ*
→ 𝐴 ≤ 𝐴) | 
| 38 | 37 | ad3antrrr 730 | . . . . . . . 8
⊢ ((((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) ∧ 𝐶 ∈ ℝ*) ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐴) | 
| 39 |  | breq2 5146 | . . . . . . . 8
⊢ (𝐶 = 𝐴 → (𝐴 ≤ 𝐶 ↔ 𝐴 ≤ 𝐴)) | 
| 40 | 38, 39 | syl5ibrcom 247 | . . . . . . 7
⊢ ((((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) ∧ 𝐶 ∈ ℝ*) ∧ 𝐴 ≤ 𝐵) → (𝐶 = 𝐴 → 𝐴 ≤ 𝐶)) | 
| 41 |  | xrltle 13192 | . . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ*
∧ 𝐶 ∈
ℝ*) → (𝐴 < 𝐶 → 𝐴 ≤ 𝐶)) | 
| 42 | 41 | adantr 480 | . . . . . . . . 9
⊢ (((𝐴 ∈ ℝ*
∧ 𝐶 ∈
ℝ*) ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐶 → 𝐴 ≤ 𝐶)) | 
| 43 | 42 | adantllr 719 | . . . . . . . 8
⊢ ((((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) ∧ 𝐶 ∈ ℝ*) ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐶 → 𝐴 ≤ 𝐶)) | 
| 44 | 43 | adantrd 491 | . . . . . . 7
⊢ ((((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) ∧ 𝐶 ∈ ℝ*) ∧ 𝐴 ≤ 𝐵) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐴 ≤ 𝐶)) | 
| 45 |  | simpr 484 | . . . . . . . 8
⊢ ((((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) ∧ 𝐶 ∈ ℝ*) ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵) | 
| 46 |  | breq2 5146 | . . . . . . . 8
⊢ (𝐶 = 𝐵 → (𝐴 ≤ 𝐶 ↔ 𝐴 ≤ 𝐵)) | 
| 47 | 45, 46 | syl5ibrcom 247 | . . . . . . 7
⊢ ((((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) ∧ 𝐶 ∈ ℝ*) ∧ 𝐴 ≤ 𝐵) → (𝐶 = 𝐵 → 𝐴 ≤ 𝐶)) | 
| 48 | 40, 44, 47 | 3jaod 1430 | . . . . . 6
⊢ ((((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) ∧ 𝐶 ∈ ℝ*) ∧ 𝐴 ≤ 𝐵) → ((𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵) → 𝐴 ≤ 𝐶)) | 
| 49 | 48 | exp31 419 | . . . . 5
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → (𝐶 ∈ ℝ* → (𝐴 ≤ 𝐵 → ((𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵) → 𝐴 ≤ 𝐶)))) | 
| 50 | 49 | 3impd 1348 | . . . 4
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)) → 𝐴 ≤ 𝐶)) | 
| 51 |  | breq1 5145 | . . . . . . . 8
⊢ (𝐶 = 𝐴 → (𝐶 ≤ 𝐵 ↔ 𝐴 ≤ 𝐵)) | 
| 52 | 45, 51 | syl5ibrcom 247 | . . . . . . 7
⊢ ((((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) ∧ 𝐶 ∈ ℝ*) ∧ 𝐴 ≤ 𝐵) → (𝐶 = 𝐴 → 𝐶 ≤ 𝐵)) | 
| 53 |  | xrltle 13192 | . . . . . . . . . . 11
⊢ ((𝐶 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → (𝐶 < 𝐵 → 𝐶 ≤ 𝐵)) | 
| 54 | 53 | ancoms 458 | . . . . . . . . . 10
⊢ ((𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ*) → (𝐶 < 𝐵 → 𝐶 ≤ 𝐵)) | 
| 55 | 54 | adantld 490 | . . . . . . . . 9
⊢ ((𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ*) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐶 ≤ 𝐵)) | 
| 56 | 55 | adantll 714 | . . . . . . . 8
⊢ (((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) ∧ 𝐶 ∈ ℝ*) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐶 ≤ 𝐵)) | 
| 57 | 56 | adantr 480 | . . . . . . 7
⊢ ((((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) ∧ 𝐶 ∈ ℝ*) ∧ 𝐴 ≤ 𝐵) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐶 ≤ 𝐵)) | 
| 58 |  | xrleid 13194 | . . . . . . . . 9
⊢ (𝐵 ∈ ℝ*
→ 𝐵 ≤ 𝐵) | 
| 59 | 58 | ad3antlr 731 | . . . . . . . 8
⊢ ((((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) ∧ 𝐶 ∈ ℝ*) ∧ 𝐴 ≤ 𝐵) → 𝐵 ≤ 𝐵) | 
| 60 |  | breq1 5145 | . . . . . . . 8
⊢ (𝐶 = 𝐵 → (𝐶 ≤ 𝐵 ↔ 𝐵 ≤ 𝐵)) | 
| 61 | 59, 60 | syl5ibrcom 247 | . . . . . . 7
⊢ ((((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) ∧ 𝐶 ∈ ℝ*) ∧ 𝐴 ≤ 𝐵) → (𝐶 = 𝐵 → 𝐶 ≤ 𝐵)) | 
| 62 | 52, 57, 61 | 3jaod 1430 | . . . . . 6
⊢ ((((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) ∧ 𝐶 ∈ ℝ*) ∧ 𝐴 ≤ 𝐵) → ((𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵) → 𝐶 ≤ 𝐵)) | 
| 63 | 62 | exp31 419 | . . . . 5
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → (𝐶 ∈ ℝ* → (𝐴 ≤ 𝐵 → ((𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵) → 𝐶 ≤ 𝐵)))) | 
| 64 | 63 | 3impd 1348 | . . . 4
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)) → 𝐶 ≤ 𝐵)) | 
| 65 | 36, 50, 64 | 3jcad 1129 | . . 3
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)) → (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | 
| 66 | 34, 65 | impbid 212 | . 2
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)))) | 
| 67 | 1, 66 | bitrd 279 | 1
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ∨ 𝐶 = 𝐵)))) |