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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 6789 | . . 3 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
2 | pinn 6789 | . . 3 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω) | |
3 | nntri3or 6189 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)) | |
4 | 1, 2, 3 | syl2an 283 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)) |
5 | ltpiord 6799 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
6 | biidd 170 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 = 𝐵 ↔ 𝐴 = 𝐵)) | |
7 | ltpiord 6799 | . . . 4 ⊢ ((𝐵 ∈ N ∧ 𝐴 ∈ N) → (𝐵 <N 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
8 | 7 | ancoms 264 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐵 <N 𝐴 ↔ 𝐵 ∈ 𝐴)) |
9 | 5, 6, 8 | 3orbi123d 1245 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ((𝐴 <N 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 <N 𝐴) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))) |
10 | 4, 9 | mpbird 165 | 1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 <N 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∨ w3o 921 = wceq 1287 ∈ wcel 1436 class class class wbr 3814 ωcom 4371 Ncnpi 6752 <N clti 6755 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-sep 3925 ax-nul 3933 ax-pow 3977 ax-pr 4003 ax-un 4227 ax-iinf 4369 |
This theorem depends on definitions: df-bi 115 df-3or 923 df-3an 924 df-tru 1290 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ral 2360 df-rex 2361 df-v 2616 df-dif 2988 df-un 2990 df-in 2992 df-ss 2999 df-nul 3273 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-uni 3631 df-int 3666 df-br 3815 df-opab 3869 df-tr 3905 df-eprel 4083 df-iord 4160 df-on 4162 df-suc 4165 df-iom 4372 df-xp 4410 df-ni 6784 df-lti 6787 |
This theorem is referenced by: nqtri3or 6876 caucvgprlemnkj 7146 caucvgprlemnbj 7147 caucvgprprlemnkj 7172 caucvgprprlemnbj 7173 caucvgsr 7268 |
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