ltrennb - Intuitionistic Logic Explorer

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Theorem ltrennb 7795

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 7364	. . 3 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (𝐽 <_N 𝐾 ↔ [⟨𝐽, 1_o⟩] ~_Q <_Q [⟨𝐾, 1_o⟩] ~_Q ))
2		nnnq 7363	. . . . 5 ⊢ (𝐽 ∈ N → [⟨𝐽, 1_o⟩] ~_Q ∈ Q)
3	2	adantr 274	. . . 4 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → [⟨𝐽, 1_o⟩] ~_Q ∈ Q)
4		nnnq 7363	. . . . 5 ⊢ (𝐾 ∈ N → [⟨𝐾, 1_o⟩] ~_Q ∈ Q)
5	4	adantl 275	. . . 4 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → [⟨𝐾, 1_o⟩] ~_Q ∈ Q)
6		ltnqpr 7534	. . . 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 7488	. . . . 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 7488	. . . . 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 7728	. . . 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 7780	. 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 2136 {cab 2151 ⟨cop 3579 class class class wbr 3982 (class class class)co 5842 1_oc1o 6377 [cec 6499 Ncnpi 7213 <_N clti 7216 ~_Q ceq 7220 Qcnq 7221 <_Q cltq 7226 Pcnp 7232 1_Pc1p 7233 +_P cpp 7234 <_P cltp 7236 ~_R cer 7237 0_Rc0r 7239 <_R cltr 7244 <_ℝ cltrr 7757

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 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-eprel 4267 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-irdg 6338 df-1o 6384 df-2o 6385 df-oadd 6388 df-omul 6389 df-er 6501 df-ec 6503 df-qs 6507 df-ni 7245 df-pli 7246 df-mi 7247 df-lti 7248 df-plpq 7285 df-mpq 7286 df-enq 7288 df-nqqs 7289 df-plqqs 7290 df-mqqs 7291 df-1nqqs 7292 df-rq 7293 df-ltnqqs 7294 df-enq0 7365 df-nq0 7366 df-0nq0 7367 df-plq0 7368 df-mq0 7369 df-inp 7407 df-i1p 7408 df-iplp 7409 df-iltp 7411 df-enr 7667 df-nr 7668 df-ltr 7671 df-0r 7672 df-r 7763 df-lt 7766

This theorem is referenced by: ltrenn 7796 axcaucvglemres 7840

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