ltrennb - Intuitionistic Logic Explorer

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Theorem ltrennb 7938

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 7507	. . 3 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (𝐽 <_N 𝐾 ↔ [⟨𝐽, 1_o⟩] ~_Q <_Q [⟨𝐾, 1_o⟩] ~_Q ))
2		nnnq 7506	. . . . 5 ⊢ (𝐽 ∈ N → [⟨𝐽, 1_o⟩] ~_Q ∈ Q)
3	2	adantr 276	. . . 4 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → [⟨𝐽, 1_o⟩] ~_Q ∈ Q)
4		nnnq 7506	. . . . 5 ⊢ (𝐾 ∈ N → [⟨𝐾, 1_o⟩] ~_Q ∈ Q)
5	4	adantl 277	. . . 4 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → [⟨𝐾, 1_o⟩] ~_Q ∈ Q)
6		ltnqpr 7677	. . . 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 7631	. . . . 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 7631	. . . . 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 7871	. . . 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 7923	. 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 2167 {cab 2182 ⟨cop 3626 class class class wbr 4034 (class class class)co 5925 1_oc1o 6476 [cec 6599 Ncnpi 7356 <_N clti 7359 ~_Q ceq 7363 Qcnq 7364 <_Q cltq 7369 Pcnp 7375 1_Pc1p 7376 +_P cpp 7377 <_P cltp 7379 ~_R cer 7380 0_Rc0r 7382 <_R cltr 7387 <_ℝ cltrr 7900

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-eprel 4325 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-recs 6372 df-irdg 6437 df-1o 6483 df-2o 6484 df-oadd 6487 df-omul 6488 df-er 6601 df-ec 6603 df-qs 6607 df-ni 7388 df-pli 7389 df-mi 7390 df-lti 7391 df-plpq 7428 df-mpq 7429 df-enq 7431 df-nqqs 7432 df-plqqs 7433 df-mqqs 7434 df-1nqqs 7435 df-rq 7436 df-ltnqqs 7437 df-enq0 7508 df-nq0 7509 df-0nq0 7510 df-plq0 7511 df-mq0 7512 df-inp 7550 df-i1p 7551 df-iplp 7552 df-iltp 7554 df-enr 7810 df-nr 7811 df-ltr 7814 df-0r 7815 df-r 7906 df-lt 7909

This theorem is referenced by: ltrenn 7939 axcaucvglemres 7983

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