zdclt - Intuitionistic Logic Explorer

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Theorem zdclt 9727

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 9692	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴))
2		zre 9653	. . 3 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
3		zre 9653	. . 3 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ)
4		orc 724	. . . . . 6 ⊢ (𝐴 < 𝐵 → (𝐴 < 𝐵 ∨ ¬ 𝐴 < 𝐵))
5		df-dc 847	. . . . . 6 ⊢ (DECID 𝐴 < 𝐵 ↔ (𝐴 < 𝐵 ∨ ¬ 𝐴 < 𝐵))
6	4, 5	sylibr 134	. . . . 5 ⊢ (𝐴 < 𝐵 → DECID 𝐴 < 𝐵)
7	6	a1i 9	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → DECID 𝐴 < 𝐵))
8		ltnr 8403	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
9	8	adantr 276	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → ¬ 𝐴 < 𝐴)
10		breq2 4134	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (𝐴 < 𝐴 ↔ 𝐴 < 𝐵))
11	10	adantl 277	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → (𝐴 < 𝐴 ↔ 𝐴 < 𝐵))
12	9, 11	mtbid 683	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → ¬ 𝐴 < 𝐵)
13		olc 723	. . . . . . . 8 ⊢ (¬ 𝐴 < 𝐵 → (𝐴 < 𝐵 ∨ ¬ 𝐴 < 𝐵))
14	13, 5	sylibr 134	. . . . . . 7 ⊢ (¬ 𝐴 < 𝐵 → DECID 𝐴 < 𝐵)
15	12, 14	syl 14	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → DECID 𝐴 < 𝐵)
16	15	ex 115	. . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 = 𝐵 → DECID 𝐴 < 𝐵))
17	16	adantr 276	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 → DECID 𝐴 < 𝐵))
18		ltnsym 8412	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 → ¬ 𝐴 < 𝐵))
19	18	ancoms 268	. . . . 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 1345	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴) → DECID 𝐴 < 𝐵))
22	2, 3, 21	syl2an 289	. 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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 720 DECID wdc 846 ∨ w3o 1008 = wceq 1402 ∈ wcel 2209 class class class wbr 4130 ℝcr 8179 < clt 8361 ℤcz 9649

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4249 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-cnex 8271 ax-resscn 8272 ax-1cn 8273 ax-1re 8274 ax-icn 8275 ax-addcl 8276 ax-addrcl 8277 ax-mulcl 8278 ax-addcom 8280 ax-addass 8282 ax-distr 8284 ax-i2m1 8285 ax-0lt1 8286 ax-0id 8288 ax-rnegex 8289 ax-cnre 8291 ax-pre-ltirr 8292 ax-pre-ltwlin 8293 ax-pre-lttrn 8294 ax-pre-ltadd 8296

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-br 4131 df-opab 4193 df-id 4438 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-iota 5337 df-fun 5379 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-pnf 8363 df-mnf 8364 df-xr 8365 df-ltxr 8366 df-le 8367 df-sub 8501 df-neg 8502 df-inn 9308 df-n0 9569 df-z 9650

This theorem is used by: fztri3or 10454 modifeq2int 10838 modsumfzodifsn 10848 seqf1oglem1 10971 seqf1oglem2 10972 exp3val 10993 ccatsymb 11386 fzowrddc 11435 swrd0g 11448 cvgratz 12318 bitsfzolem 12740 bitsmod 12742 infpnlem1 13161 infpnlem2 13162 ballotfilemefi 13289 gzsumfzval 13764 gzsumcl 13857 mulgval 13978 mulgfng 13980 subgmulg 14044 gzsumreidx 14225 gzsumsubmcl 14226 gzsummhm 14229 lgsval 16289 lgscllem 16292 lgsneg 16309 lgsdilem 16312 lgsdir 16320 lgsdi 16322 lgsne0 16323 lgsquadlemsfi 16360 lgsquadlem3 16364

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