zdclt - Intuitionistic Logic Explorer

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Theorem zdclt 8719

Description: Integer < is decidable. (Contributed by Jim Kingdon, 1-Jun-2020.)

**Assertion**
Ref	Expression
zdclt	⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 < 𝐵)

**Proof of Theorem zdclt**
Step	Hyp	Ref	Expression
1		ztri3or 8688	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴))
2		zre 8649	. . 3 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
3		zre 8649	. . 3 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ)
4		orc 666	. . . . . 6 ⊢ (𝐴 < 𝐵 → (𝐴 < 𝐵 ∨ ¬ 𝐴 < 𝐵))
5		df-dc 777	. . . . . 6 ⊢ (DECID 𝐴 < 𝐵 ↔ (𝐴 < 𝐵 ∨ ¬ 𝐴 < 𝐵))
6	4, 5	sylibr 132	. . . . 5 ⊢ (𝐴 < 𝐵 → DECID 𝐴 < 𝐵)
7	6	a1i 9	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → DECID 𝐴 < 𝐵))
8		ltnr 7464	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
9	8	adantr 270	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → ¬ 𝐴 < 𝐴)
10		breq2 3815	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (𝐴 < 𝐴 ↔ 𝐴 < 𝐵))
11	10	adantl 271	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → (𝐴 < 𝐴 ↔ 𝐴 < 𝐵))
12	9, 11	mtbid 630	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → ¬ 𝐴 < 𝐵)
13		olc 665	. . . . . . . 8 ⊢ (¬ 𝐴 < 𝐵 → (𝐴 < 𝐵 ∨ ¬ 𝐴 < 𝐵))
14	13, 5	sylibr 132	. . . . . . 7 ⊢ (¬ 𝐴 < 𝐵 → DECID 𝐴 < 𝐵)
15	12, 14	syl 14	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → DECID 𝐴 < 𝐵)
16	15	ex 113	. . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 = 𝐵 → DECID 𝐴 < 𝐵))
17	16	adantr 270	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 → DECID 𝐴 < 𝐵))
18		ltnsym 7473	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 → ¬ 𝐴 < 𝐵))
19	18	ancoms 264	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴 → ¬ 𝐴 < 𝐵))
20	19, 14	syl6 33	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴 → DECID 𝐴 < 𝐵))
21	7, 17, 20	3jaod 1236	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴) → DECID 𝐴 < 𝐵))
22	2, 3, 21	syl2an 283	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴) → DECID 𝐴 < 𝐵))
23	1, 22	mpd 13	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 < 𝐵)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 ∨ wo 662 DECID wdc 776 ∨ w3o 919 = wceq 1285 ∈ wcel 1434 class class class wbr 3811 ℝcr 7251 < clt 7424 ℤcz 8645

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 3999 ax-un 4223 ax-setind 4315 ax-cnex 7338 ax-resscn 7339 ax-1cn 7340 ax-1re 7341 ax-icn 7342 ax-addcl 7343 ax-addrcl 7344 ax-mulcl 7345 ax-addcom 7347 ax-addass 7349 ax-distr 7351 ax-i2m1 7352 ax-0lt1 7353 ax-0id 7355 ax-rnegex 7356 ax-cnre 7358 ax-pre-ltirr 7359 ax-pre-ltwlin 7360 ax-pre-lttrn 7361 ax-pre-ltadd 7363

This theorem depends on definitions: df-bi 115 df-dc 777 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 4083 df-xp 4406 df-rel 4407 df-cnv 4408 df-co 4409 df-dm 4410 df-iota 4933 df-fun 4970 df-fv 4976 df-riota 5546 df-ov 5593 df-oprab 5594 df-mpt2 5595 df-pnf 7426 df-mnf 7427 df-xr 7428 df-ltxr 7429 df-le 7430 df-sub 7557 df-neg 7558 df-inn 8316 df-n0 8565 df-z 8646

This theorem is referenced by: fztri3or 9347 modifeq2int 9681 modsumfzodifsn 9691 expival 9793

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