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Mirrors > Home > MPE Home > Th. List > ordtri3 | Structured version Visualization version GIF version |
Description: A trichotomy law for ordinals. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof shortened by JJ, 24-Sep-2021.) |
Ref | Expression |
---|---|
ordtri3 | ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordirr 6203 | . . . . . 6 ⊢ (Ord 𝐵 → ¬ 𝐵 ∈ 𝐵) | |
2 | 1 | adantl 484 | . . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ¬ 𝐵 ∈ 𝐵) |
3 | eleq2 2901 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ 𝐵)) | |
4 | 3 | notbid 320 | . . . . 5 ⊢ (𝐴 = 𝐵 → (¬ 𝐵 ∈ 𝐴 ↔ ¬ 𝐵 ∈ 𝐵)) |
5 | 2, 4 | syl5ibrcom 249 | . . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 → ¬ 𝐵 ∈ 𝐴)) |
6 | 5 | pm4.71d 564 | . . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∧ ¬ 𝐵 ∈ 𝐴))) |
7 | pm5.61 997 | . . . 4 ⊢ (((𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ∧ ¬ 𝐵 ∈ 𝐴) ↔ (𝐴 = 𝐵 ∧ ¬ 𝐵 ∈ 𝐴)) | |
8 | pm4.52 981 | . . . 4 ⊢ (((𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ∧ ¬ 𝐵 ∈ 𝐴) ↔ ¬ (¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ∨ 𝐵 ∈ 𝐴)) | |
9 | 7, 8 | bitr3i 279 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ ¬ 𝐵 ∈ 𝐴) ↔ ¬ (¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ∨ 𝐵 ∈ 𝐴)) |
10 | 6, 9 | syl6bb 289 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ∨ 𝐵 ∈ 𝐴))) |
11 | ordtri2 6220 | . . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))) | |
12 | 11 | orbi1d 913 | . . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴) ↔ (¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ∨ 𝐵 ∈ 𝐴))) |
13 | 12 | notbid 320 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴) ↔ ¬ (¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ∨ 𝐵 ∈ 𝐴))) |
14 | 10, 13 | bitr4d 284 | 1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 Ord word 6184 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-tr 5165 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-ord 6188 |
This theorem is referenced by: ordunisuc2 7553 tz7.48lem 8071 oacan 8168 omcan 8189 oecan 8209 omsmo 8275 omopthi 8278 inf3lem6 9090 cantnfp1lem3 9137 infpssrlem5 9723 fin23lem24 9738 isf32lem4 9772 om2uzf1oi 13315 |
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