zdclt - Intuitionistic Logic Explorer

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Theorem zdclt 9276

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 9242	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴))
2		zre 9203	. . 3 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
3		zre 9203	. . 3 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ)
4		orc 707	. . . . . 6 ⊢ (𝐴 < 𝐵 → (𝐴 < 𝐵 ∨ ¬ 𝐴 < 𝐵))
5		df-dc 830	. . . . . 6 ⊢ (DECID 𝐴 < 𝐵 ↔ (𝐴 < 𝐵 ∨ ¬ 𝐴 < 𝐵))
6	4, 5	sylibr 133	. . . . 5 ⊢ (𝐴 < 𝐵 → DECID 𝐴 < 𝐵)
7	6	a1i 9	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → DECID 𝐴 < 𝐵))
8		ltnr 7983	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
9	8	adantr 274	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → ¬ 𝐴 < 𝐴)
10		breq2 3991	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (𝐴 < 𝐴 ↔ 𝐴 < 𝐵))
11	10	adantl 275	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → (𝐴 < 𝐴 ↔ 𝐴 < 𝐵))
12	9, 11	mtbid 667	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → ¬ 𝐴 < 𝐵)
13		olc 706	. . . . . . . 8 ⊢ (¬ 𝐴 < 𝐵 → (𝐴 < 𝐵 ∨ ¬ 𝐴 < 𝐵))
14	13, 5	sylibr 133	. . . . . . 7 ⊢ (¬ 𝐴 < 𝐵 → DECID 𝐴 < 𝐵)
15	12, 14	syl 14	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → DECID 𝐴 < 𝐵)
16	15	ex 114	. . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 = 𝐵 → DECID 𝐴 < 𝐵))
17	16	adantr 274	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 → DECID 𝐴 < 𝐵))
18		ltnsym 7992	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 → ¬ 𝐴 < 𝐵))
19	18	ancoms 266	. . . . 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 1299	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴) → DECID 𝐴 < 𝐵))
22	2, 3, 21	syl2an 287	. 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 103 ↔ wb 104 ∨ wo 703 DECID wdc 829 ∨ w3o 972 = wceq 1348 ∈ wcel 2141 class class class wbr 3987 ℝcr 7760 < clt 7941 ℤcz 9199

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-addcom 7861 ax-addass 7863 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-0id 7869 ax-rnegex 7870 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-ltadd 7877

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-inn 8866 df-n0 9123 df-z 9200

This theorem is referenced by: fztri3or 9982 modifeq2int 10329 modsumfzodifsn 10339 exp3val 10465 cvgratz 11482 infpnlem1 12298 infpnlem2 12299 lgsval 13620 lgscllem 13623 lgsneg 13640 lgsdilem 13643 lgsdir 13651 lgsdi 13653 lgsne0 13654

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