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Mirrors > Home > ILE Home > Th. List > pitri3or | GIF version |
Description: Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.) |
Ref | Expression |
---|---|
pitri3or | ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 <N 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pinn 7018 | . . 3 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
2 | pinn 7018 | . . 3 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω) | |
3 | nntri3or 6319 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)) | |
4 | 1, 2, 3 | syl2an 285 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)) |
5 | ltpiord 7028 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
6 | biidd 171 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 = 𝐵 ↔ 𝐴 = 𝐵)) | |
7 | ltpiord 7028 | . . . 4 ⊢ ((𝐵 ∈ N ∧ 𝐴 ∈ N) → (𝐵 <N 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
8 | 7 | ancoms 266 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐵 <N 𝐴 ↔ 𝐵 ∈ 𝐴)) |
9 | 5, 6, 8 | 3orbi123d 1257 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ((𝐴 <N 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 <N 𝐴) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))) |
10 | 4, 9 | mpbird 166 | 1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 <N 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∨ w3o 929 = wceq 1299 ∈ wcel 1448 class class class wbr 3875 ωcom 4442 Ncnpi 6981 <N clti 6984 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-iinf 4440 |
This theorem depends on definitions: df-bi 116 df-3or 931 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-br 3876 df-opab 3930 df-tr 3967 df-eprel 4149 df-iord 4226 df-on 4228 df-suc 4231 df-iom 4443 df-xp 4483 df-ni 7013 df-lti 7016 |
This theorem is referenced by: nqtri3or 7105 caucvgprlemnkj 7375 caucvgprlemnbj 7376 caucvgprprlemnkj 7401 caucvgprprlemnbj 7402 caucvgsr 7497 |
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