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| Mirrors > Home > MPE Home > Th. List > ordtri1 | Structured version Visualization version GIF version | ||
| Description: A trichotomy law for ordinals. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
| Ref | Expression |
|---|---|
| ordtri1 | ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ordsseleq 6413 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) | |
| 2 | ordn2lp 6404 | . . . . 5 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)) | |
| 3 | imnan 399 | . . . . 5 ⊢ ((𝐴 ∈ 𝐵 → ¬ 𝐵 ∈ 𝐴) ↔ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)) | |
| 4 | 2, 3 | sylibr 234 | . . . 4 ⊢ (Ord 𝐴 → (𝐴 ∈ 𝐵 → ¬ 𝐵 ∈ 𝐴)) |
| 5 | ordirr 6402 | . . . . 5 ⊢ (Ord 𝐵 → ¬ 𝐵 ∈ 𝐵) | |
| 6 | eleq2 2830 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ 𝐵)) | |
| 7 | 6 | notbid 318 | . . . . 5 ⊢ (𝐴 = 𝐵 → (¬ 𝐵 ∈ 𝐴 ↔ ¬ 𝐵 ∈ 𝐵)) |
| 8 | 5, 7 | syl5ibrcom 247 | . . . 4 ⊢ (Ord 𝐵 → (𝐴 = 𝐵 → ¬ 𝐵 ∈ 𝐴)) |
| 9 | 4, 8 | jaao 957 | . . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) → ¬ 𝐵 ∈ 𝐴)) |
| 10 | ordtri3or 6416 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)) | |
| 11 | df-3or 1088 | . . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ↔ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ∨ 𝐵 ∈ 𝐴)) | |
| 12 | 10, 11 | sylib 218 | . . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ∨ 𝐵 ∈ 𝐴)) |
| 13 | 12 | orcomd 872 | . . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵 ∈ 𝐴 ∨ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) |
| 14 | 13 | ord 865 | . . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ 𝐵 ∈ 𝐴 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) |
| 15 | 9, 14 | impbid 212 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ↔ ¬ 𝐵 ∈ 𝐴)) |
| 16 | 1, 15 | bitrd 279 | 1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∨ w3o 1086 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 Ord word 6383 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 |
| This theorem is referenced by: ontri1 6418 ordtri2 6419 ordtri4 6421 ordtr3 6429 ordintdif 6434 ordtri2or 6482 ordsucss 7838 ordsucsssuc 7843 ordsucuniel 7844 limsssuc 7871 ssnlim 7907 smoword 8406 tfrlem15 8432 nnaword 8665 nnawordex 8675 eldifsucnn 8702 nndomog 9253 nndomogOLD 9263 onomeneq 9265 onomeneqOLD 9266 isfinite2 9334 unfilem1 9343 wofib 9585 cantnflem1 9729 ttrcltr 9756 dmttrcl 9761 alephgeom 10122 alephdom2 10127 cflim2 10303 fin67 10435 winainflem 10733 finminlem 36319 ordeldif 43271 ordeldifsucon 43272 ordeldif1o 43273 |
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