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Mirrors > Home > ILE Home > Th. List > pitric | GIF version |
Description: Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.) |
Ref | Expression |
---|---|
pitric | ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <N 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pinn 7085 | . . 3 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
2 | pinn 7085 | . . 3 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω) | |
3 | nntri2 6358 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))) | |
4 | 1, 2, 3 | syl2an 287 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))) |
5 | ltpiord 7095 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
6 | ltpiord 7095 | . . . . 5 ⊢ ((𝐵 ∈ N ∧ 𝐴 ∈ N) → (𝐵 <N 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
7 | 6 | ancoms 266 | . . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐵 <N 𝐴 ↔ 𝐵 ∈ 𝐴)) |
8 | 7 | orbi2d 764 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ((𝐴 = 𝐵 ∨ 𝐵 <N 𝐴) ↔ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))) |
9 | 8 | notbid 641 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (¬ (𝐴 = 𝐵 ∨ 𝐵 <N 𝐴) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))) |
10 | 4, 5, 9 | 3bitr4d 219 | 1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <N 𝐴))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 682 = wceq 1316 ∈ wcel 1465 class class class wbr 3899 ωcom 4474 Ncnpi 7048 <N clti 7051 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-tr 3997 df-eprel 4181 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-ni 7080 df-lti 7083 |
This theorem is referenced by: (None) |
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