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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 7286 | . . 3 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
2 | pinn 7286 | . . 3 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω) | |
3 | nntri3or 6487 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)) | |
4 | 1, 2, 3 | syl2an 289 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)) |
5 | ltpiord 7296 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
6 | biidd 172 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 = 𝐵 ↔ 𝐴 = 𝐵)) | |
7 | ltpiord 7296 | . . . 4 ⊢ ((𝐵 ∈ N ∧ 𝐴 ∈ N) → (𝐵 <N 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
8 | 7 | ancoms 268 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐵 <N 𝐴 ↔ 𝐵 ∈ 𝐴)) |
9 | 5, 6, 8 | 3orbi123d 1311 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ((𝐴 <N 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 <N 𝐴) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))) |
10 | 4, 9 | mpbird 167 | 1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 <N 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ w3o 977 = wceq 1353 ∈ wcel 2148 class class class wbr 4000 ωcom 4585 Ncnpi 7249 <N clti 7252 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-iinf 4583 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-tr 4099 df-eprel 4285 df-iord 4362 df-on 4364 df-suc 4367 df-iom 4586 df-xp 4628 df-ni 7281 df-lti 7284 |
This theorem is referenced by: nqtri3or 7373 caucvgprlemnkj 7643 caucvgprlemnbj 7644 caucvgprprlemnkj 7669 caucvgprprlemnbj 7670 caucvgsr 7779 |
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