ltrennb - Intuitionistic Logic Explorer

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Theorem ltrennb 7816

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 7385	. . 3 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (𝐽 <_N 𝐾 ↔ [⟨𝐽, 1_o⟩] ~_Q <_Q [⟨𝐾, 1_o⟩] ~_Q ))
2		nnnq 7384	. . . . 5 ⊢ (𝐽 ∈ N → [⟨𝐽, 1_o⟩] ~_Q ∈ Q)
3	2	adantr 274	. . . 4 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → [⟨𝐽, 1_o⟩] ~_Q ∈ Q)
4		nnnq 7384	. . . . 5 ⊢ (𝐾 ∈ N → [⟨𝐾, 1_o⟩] ~_Q ∈ Q)
5	4	adantl 275	. . . 4 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → [⟨𝐾, 1_o⟩] ~_Q ∈ Q)
6		ltnqpr 7555	. . . 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 409	. . 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 7509	. . . . 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 7509	. . . . 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 7749	. . . 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 409	. . 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 213	. 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 7801	. 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 197	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 103 ↔ wb 104 ∈ wcel 2141 {cab 2156 ⟨cop 3586 class class class wbr 3989 (class class class)co 5853 1_oc1o 6388 [cec 6511 Ncnpi 7234 <_N clti 7237 ~_Q ceq 7241 Qcnq 7242 <_Q cltq 7247 Pcnp 7253 1_Pc1p 7254 +_P cpp 7255 <_P cltp 7257 ~_R cer 7258 0_Rc0r 7260 <_R cltr 7265 <_ℝ cltrr 7778

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-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572

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-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-eprel 4274 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-1o 6395 df-2o 6396 df-oadd 6399 df-omul 6400 df-er 6513 df-ec 6515 df-qs 6519 df-ni 7266 df-pli 7267 df-mi 7268 df-lti 7269 df-plpq 7306 df-mpq 7307 df-enq 7309 df-nqqs 7310 df-plqqs 7311 df-mqqs 7312 df-1nqqs 7313 df-rq 7314 df-ltnqqs 7315 df-enq0 7386 df-nq0 7387 df-0nq0 7388 df-plq0 7389 df-mq0 7390 df-inp 7428 df-i1p 7429 df-iplp 7430 df-iltp 7432 df-enr 7688 df-nr 7689 df-ltr 7692 df-0r 7693 df-r 7784 df-lt 7787

This theorem is referenced by: ltrenn 7817 axcaucvglemres 7861

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