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Mirrors > Home > ILE Home > Th. List > zletric | GIF version |
Description: Trichotomy law. (Contributed by Jim Kingdon, 27-Mar-2020.) |
Ref | Expression |
---|---|
zletric | ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 8650 | . . 3 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
2 | zre 8650 | . . 3 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ) | |
3 | 1, 2 | anim12i 331 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) |
4 | ztri3or 8689 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴)) | |
5 | ltle 7475 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) | |
6 | orc 666 | . . . 4 ⊢ (𝐴 ≤ 𝐵 → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) | |
7 | 5, 6 | syl6 33 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))) |
8 | eqle 7479 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → 𝐴 ≤ 𝐵) | |
9 | 8 | ex 113 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 = 𝐵 → 𝐴 ≤ 𝐵)) |
10 | 9 | adantr 270 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 → 𝐴 ≤ 𝐵)) |
11 | 10, 6 | syl6 33 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))) |
12 | ltle 7475 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 → 𝐵 ≤ 𝐴)) | |
13 | 12 | ancoms 264 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴 → 𝐵 ≤ 𝐴)) |
14 | olc 665 | . . . 4 ⊢ (𝐵 ≤ 𝐴 → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) | |
15 | 13, 14 | syl6 33 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴 → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))) |
16 | 7, 11, 15 | 3jaod 1236 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))) |
17 | 3, 4, 16 | sylc 61 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∨ wo 662 ∨ w3o 919 = wceq 1285 ∈ wcel 1434 class class class wbr 3811 ℝcr 7252 < clt 7425 ≤ cle 7426 ℤcz 8646 |
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 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 4000 ax-un 4224 ax-setind 4316 ax-cnex 7339 ax-resscn 7340 ax-1cn 7341 ax-1re 7342 ax-icn 7343 ax-addcl 7344 ax-addrcl 7345 ax-mulcl 7346 ax-addcom 7348 ax-addass 7350 ax-distr 7352 ax-i2m1 7353 ax-0lt1 7354 ax-0id 7356 ax-rnegex 7357 ax-cnre 7359 ax-pre-ltirr 7360 ax-pre-ltwlin 7361 ax-pre-lttrn 7362 ax-pre-ltadd 7364 |
This theorem depends on definitions: df-bi 115 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2614 df-sbc 2827 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-br 3812 df-opab 3866 df-id 4084 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-iota 4934 df-fun 4971 df-fv 4977 df-riota 5547 df-ov 5594 df-oprab 5595 df-mpt2 5596 df-pnf 7427 df-mnf 7428 df-xr 7429 df-ltxr 7430 df-le 7431 df-sub 7558 df-neg 7559 df-inn 8317 df-n0 8566 df-z 8647 |
This theorem is referenced by: elz2 8714 uztric 8935 z2ge 9183 nn0abscl 10345 fzomaxdif 10373 |
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