Proof of Theorem smoord
| Step | Hyp | Ref
| Expression |
| 1 | | smodm2 8395 |
. . 3
⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → Ord 𝐴) |
| 2 | | simprl 771 |
. . 3
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐶 ∈ 𝐴) |
| 3 | | ordelord 6406 |
. . 3
⊢ ((Ord
𝐴 ∧ 𝐶 ∈ 𝐴) → Ord 𝐶) |
| 4 | 1, 2, 3 | syl2an2r 685 |
. 2
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → Ord 𝐶) |
| 5 | | simprr 773 |
. . 3
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐷 ∈ 𝐴) |
| 6 | | ordelord 6406 |
. . 3
⊢ ((Ord
𝐴 ∧ 𝐷 ∈ 𝐴) → Ord 𝐷) |
| 7 | 1, 5, 6 | syl2an2r 685 |
. 2
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → Ord 𝐷) |
| 8 | | ordtri3or 6416 |
. . 3
⊢ ((Ord
𝐶 ∧ Ord 𝐷) → (𝐶 ∈ 𝐷 ∨ 𝐶 = 𝐷 ∨ 𝐷 ∈ 𝐶)) |
| 9 | | simp3 1139 |
. . . . . 6
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) ∧ 𝐶 ∈ 𝐷) → 𝐶 ∈ 𝐷) |
| 10 | | smoel2 8403 |
. . . . . . . . 9
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐷)) → (𝐹‘𝐶) ∈ (𝐹‘𝐷)) |
| 11 | 10 | expr 456 |
. . . . . . . 8
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ 𝐷 ∈ 𝐴) → (𝐶 ∈ 𝐷 → (𝐹‘𝐶) ∈ (𝐹‘𝐷))) |
| 12 | 11 | adantrl 716 |
. . . . . . 7
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶 ∈ 𝐷 → (𝐹‘𝐶) ∈ (𝐹‘𝐷))) |
| 13 | 12 | 3impia 1118 |
. . . . . 6
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) ∧ 𝐶 ∈ 𝐷) → (𝐹‘𝐶) ∈ (𝐹‘𝐷)) |
| 14 | 9, 13 | 2thd 265 |
. . . . 5
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) ∧ 𝐶 ∈ 𝐷) → (𝐶 ∈ 𝐷 ↔ (𝐹‘𝐶) ∈ (𝐹‘𝐷))) |
| 15 | 14 | 3expia 1122 |
. . . 4
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶 ∈ 𝐷 → (𝐶 ∈ 𝐷 ↔ (𝐹‘𝐶) ∈ (𝐹‘𝐷)))) |
| 16 | | ordirr 6402 |
. . . . . . . . 9
⊢ (Ord
𝐶 → ¬ 𝐶 ∈ 𝐶) |
| 17 | 4, 16 | syl 17 |
. . . . . . . 8
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ 𝐶 ∈ 𝐶) |
| 18 | 17 | 3adant3 1133 |
. . . . . . 7
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) ∧ 𝐶 = 𝐷) → ¬ 𝐶 ∈ 𝐶) |
| 19 | | simp3 1139 |
. . . . . . 7
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) ∧ 𝐶 = 𝐷) → 𝐶 = 𝐷) |
| 20 | 18, 19 | neleqtrd 2863 |
. . . . . 6
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) ∧ 𝐶 = 𝐷) → ¬ 𝐶 ∈ 𝐷) |
| 21 | | smofvon2 8396 |
. . . . . . . . . 10
⊢ (Smo
𝐹 → (𝐹‘𝐶) ∈ On) |
| 22 | 21 | ad2antlr 727 |
. . . . . . . . 9
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐹‘𝐶) ∈ On) |
| 23 | | eloni 6394 |
. . . . . . . . 9
⊢ ((𝐹‘𝐶) ∈ On → Ord (𝐹‘𝐶)) |
| 24 | | ordirr 6402 |
. . . . . . . . 9
⊢ (Ord
(𝐹‘𝐶) → ¬ (𝐹‘𝐶) ∈ (𝐹‘𝐶)) |
| 25 | 22, 23, 24 | 3syl 18 |
. . . . . . . 8
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐹‘𝐶) ∈ (𝐹‘𝐶)) |
| 26 | 25 | 3adant3 1133 |
. . . . . . 7
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) ∧ 𝐶 = 𝐷) → ¬ (𝐹‘𝐶) ∈ (𝐹‘𝐶)) |
| 27 | 19 | fveq2d 6910 |
. . . . . . 7
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) ∧ 𝐶 = 𝐷) → (𝐹‘𝐶) = (𝐹‘𝐷)) |
| 28 | 26, 27 | neleqtrd 2863 |
. . . . . 6
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) ∧ 𝐶 = 𝐷) → ¬ (𝐹‘𝐶) ∈ (𝐹‘𝐷)) |
| 29 | 20, 28 | 2falsed 376 |
. . . . 5
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) ∧ 𝐶 = 𝐷) → (𝐶 ∈ 𝐷 ↔ (𝐹‘𝐶) ∈ (𝐹‘𝐷))) |
| 30 | 29 | 3expia 1122 |
. . . 4
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶 = 𝐷 → (𝐶 ∈ 𝐷 ↔ (𝐹‘𝐶) ∈ (𝐹‘𝐷)))) |
| 31 | 7 | 3adant3 1133 |
. . . . . . . 8
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) ∧ 𝐷 ∈ 𝐶) → Ord 𝐷) |
| 32 | | ordn2lp 6404 |
. . . . . . . 8
⊢ (Ord
𝐷 → ¬ (𝐷 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷)) |
| 33 | 31, 32 | syl 17 |
. . . . . . 7
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) ∧ 𝐷 ∈ 𝐶) → ¬ (𝐷 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷)) |
| 34 | | pm3.2 469 |
. . . . . . . 8
⊢ (𝐷 ∈ 𝐶 → (𝐶 ∈ 𝐷 → (𝐷 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷))) |
| 35 | 34 | 3ad2ant3 1136 |
. . . . . . 7
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) ∧ 𝐷 ∈ 𝐶) → (𝐶 ∈ 𝐷 → (𝐷 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷))) |
| 36 | 33, 35 | mtod 198 |
. . . . . 6
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) ∧ 𝐷 ∈ 𝐶) → ¬ 𝐶 ∈ 𝐷) |
| 37 | 22, 23 | syl 17 |
. . . . . . . . 9
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → Ord (𝐹‘𝐶)) |
| 38 | 37 | 3adant3 1133 |
. . . . . . . 8
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) ∧ 𝐷 ∈ 𝐶) → Ord (𝐹‘𝐶)) |
| 39 | | ordn2lp 6404 |
. . . . . . . 8
⊢ (Ord
(𝐹‘𝐶) → ¬ ((𝐹‘𝐶) ∈ (𝐹‘𝐷) ∧ (𝐹‘𝐷) ∈ (𝐹‘𝐶))) |
| 40 | 38, 39 | syl 17 |
. . . . . . 7
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) ∧ 𝐷 ∈ 𝐶) → ¬ ((𝐹‘𝐶) ∈ (𝐹‘𝐷) ∧ (𝐹‘𝐷) ∈ (𝐹‘𝐶))) |
| 41 | | smoel2 8403 |
. . . . . . . . . 10
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐶)) → (𝐹‘𝐷) ∈ (𝐹‘𝐶)) |
| 42 | 41 | adantrlr 723 |
. . . . . . . . 9
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) ∧ 𝐷 ∈ 𝐶)) → (𝐹‘𝐷) ∈ (𝐹‘𝐶)) |
| 43 | 42 | 3impb 1115 |
. . . . . . . 8
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) ∧ 𝐷 ∈ 𝐶) → (𝐹‘𝐷) ∈ (𝐹‘𝐶)) |
| 44 | | pm3.21 471 |
. . . . . . . 8
⊢ ((𝐹‘𝐷) ∈ (𝐹‘𝐶) → ((𝐹‘𝐶) ∈ (𝐹‘𝐷) → ((𝐹‘𝐶) ∈ (𝐹‘𝐷) ∧ (𝐹‘𝐷) ∈ (𝐹‘𝐶)))) |
| 45 | 43, 44 | syl 17 |
. . . . . . 7
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) ∧ 𝐷 ∈ 𝐶) → ((𝐹‘𝐶) ∈ (𝐹‘𝐷) → ((𝐹‘𝐶) ∈ (𝐹‘𝐷) ∧ (𝐹‘𝐷) ∈ (𝐹‘𝐶)))) |
| 46 | 40, 45 | mtod 198 |
. . . . . 6
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) ∧ 𝐷 ∈ 𝐶) → ¬ (𝐹‘𝐶) ∈ (𝐹‘𝐷)) |
| 47 | 36, 46 | 2falsed 376 |
. . . . 5
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) ∧ 𝐷 ∈ 𝐶) → (𝐶 ∈ 𝐷 ↔ (𝐹‘𝐶) ∈ (𝐹‘𝐷))) |
| 48 | 47 | 3expia 1122 |
. . . 4
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐷 ∈ 𝐶 → (𝐶 ∈ 𝐷 ↔ (𝐹‘𝐶) ∈ (𝐹‘𝐷)))) |
| 49 | 15, 30, 48 | 3jaod 1431 |
. . 3
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐶 ∈ 𝐷 ∨ 𝐶 = 𝐷 ∨ 𝐷 ∈ 𝐶) → (𝐶 ∈ 𝐷 ↔ (𝐹‘𝐶) ∈ (𝐹‘𝐷)))) |
| 50 | 8, 49 | syl5 34 |
. 2
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((Ord 𝐶 ∧ Ord 𝐷) → (𝐶 ∈ 𝐷 ↔ (𝐹‘𝐶) ∈ (𝐹‘𝐷)))) |
| 51 | 4, 7, 50 | mp2and 699 |
1
⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶 ∈ 𝐷 ↔ (𝐹‘𝐶) ∈ (𝐹‘𝐷))) |