ltrennb - Intuitionistic Logic Explorer

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Theorem ltrennb 7855

Description: Ordering of natural numbers with <_N or <_ℝ. (Contributed by Jim Kingdon, 13-Jul-2021.)

**Assertion**
Ref	Expression
ltrennb	⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (𝐽 <_N 𝐾 ↔ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_o⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩ <_ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_o⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩))

Distinct variable groups: 𝐽,𝑙 𝑢,𝐽 𝐾,𝑙 𝑢,𝐾

**Proof of Theorem ltrennb**
Step	Hyp	Ref	Expression
1		ltnnnq 7424	. . 3 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (𝐽 <_N 𝐾 ↔ [⟨𝐽, 1_o⟩] ~_Q <_Q [⟨𝐾, 1_o⟩] ~_Q ))
2		nnnq 7423	. . . . 5 ⊢ (𝐽 ∈ N → [⟨𝐽, 1_o⟩] ~_Q ∈ Q)
3	2	adantr 276	. . . 4 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → [⟨𝐽, 1_o⟩] ~_Q ∈ Q)
4		nnnq 7423	. . . . 5 ⊢ (𝐾 ∈ N → [⟨𝐾, 1_o⟩] ~_Q ∈ Q)
5	4	adantl 277	. . . 4 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → [⟨𝐾, 1_o⟩] ~_Q ∈ Q)
6		ltnqpr 7594	. . . 4 ⊢ (([⟨𝐽, 1_o⟩] ~_Q ∈ Q ∧ [⟨𝐾, 1_o⟩] ~_Q ∈ Q) → ([⟨𝐽, 1_o⟩] ~_Q <_Q [⟨𝐾, 1_o⟩] ~_Q ↔ ⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_o⟩] ~_Q <_Q 𝑢}⟩<_P ⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_o⟩] ~_Q <_Q 𝑢}⟩))
7	3, 5, 6	syl2anc 411	. . 3 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → ([⟨𝐽, 1_o⟩] ~_Q <_Q [⟨𝐾, 1_o⟩] ~_Q ↔ ⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_o⟩] ~_Q <_Q 𝑢}⟩<_P ⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_o⟩] ~_Q <_Q 𝑢}⟩))
8		nqprlu 7548	. . . . 5 ⊢ ([⟨𝐽, 1_o⟩] ~_Q ∈ Q → ⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_o⟩] ~_Q <_Q 𝑢}⟩ ∈ P)
9	3, 8	syl 14	. . . 4 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → ⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_o⟩] ~_Q <_Q 𝑢}⟩ ∈ P)
10		nqprlu 7548	. . . . 5 ⊢ ([⟨𝐾, 1_o⟩] ~_Q ∈ Q → ⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_o⟩] ~_Q <_Q 𝑢}⟩ ∈ P)
11	5, 10	syl 14	. . . 4 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → ⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_o⟩] ~_Q <_Q 𝑢}⟩ ∈ P)
12		prsrlt 7788	. . . 4 ⊢ ((⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_o⟩] ~_Q <_Q 𝑢}⟩ ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_o⟩] ~_Q <_Q 𝑢}⟩ ∈ P) → (⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_o⟩] ~_Q <_Q 𝑢}⟩<_P ⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_o⟩] ~_Q <_Q 𝑢}⟩ ↔ [⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_o⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R <_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_o⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R ))
13	9, 11, 12	syl2anc 411	. . 3 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_o⟩] ~_Q <_Q 𝑢}⟩<_P ⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_o⟩] ~_Q <_Q 𝑢}⟩ ↔ [⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_o⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R <_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_o⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R ))
14	1, 7, 13	3bitrd 214	. 2 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (𝐽 <_N 𝐾 ↔ [⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_o⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R <_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_o⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R ))
15		ltresr 7840	. 2 ⊢ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_o⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩ <_ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_o⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩ ↔ [⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_o⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R <_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_o⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R )
16	14, 15	bitr4di 198	1 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (𝐽 <_N 𝐾 ↔ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_o⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩ <_ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_o⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_o⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2148 {cab 2163 ⟨cop 3597 class class class wbr 4005 (class class class)co 5877 1_oc1o 6412 [cec 6535 Ncnpi 7273 <_N clti 7276 ~_Q ceq 7280 Qcnq 7281 <_Q cltq 7286 Pcnp 7292 1_Pc1p 7293 +_P cpp 7294 <_P cltp 7296 ~_R cer 7297 0_Rc0r 7299 <_R cltr 7304 <_ℝ cltrr 7817

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-eprel 4291 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-irdg 6373 df-1o 6419 df-2o 6420 df-oadd 6423 df-omul 6424 df-er 6537 df-ec 6539 df-qs 6543 df-ni 7305 df-pli 7306 df-mi 7307 df-lti 7308 df-plpq 7345 df-mpq 7346 df-enq 7348 df-nqqs 7349 df-plqqs 7350 df-mqqs 7351 df-1nqqs 7352 df-rq 7353 df-ltnqqs 7354 df-enq0 7425 df-nq0 7426 df-0nq0 7427 df-plq0 7428 df-mq0 7429 df-inp 7467 df-i1p 7468 df-iplp 7469 df-iltp 7471 df-enr 7727 df-nr 7728 df-ltr 7731 df-0r 7732 df-r 7823 df-lt 7826

This theorem is referenced by: ltrenn 7856 axcaucvglemres 7900

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