ltrennb - Intuitionistic Logic Explorer

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Theorem ltrennb 8002

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 7571	. . 3 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (𝐽 <_N 𝐾 ↔ [⟨𝐽, 1_o⟩] ~_Q <_Q [⟨𝐾, 1_o⟩] ~_Q ))
2		nnnq 7570	. . . . 5 ⊢ (𝐽 ∈ N → [⟨𝐽, 1_o⟩] ~_Q ∈ Q)
3	2	adantr 276	. . . 4 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → [⟨𝐽, 1_o⟩] ~_Q ∈ Q)
4		nnnq 7570	. . . . 5 ⊢ (𝐾 ∈ N → [⟨𝐾, 1_o⟩] ~_Q ∈ Q)
5	4	adantl 277	. . . 4 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → [⟨𝐾, 1_o⟩] ~_Q ∈ Q)
6		ltnqpr 7741	. . . 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 7695	. . . . 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 7695	. . . . 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 7935	. . . 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 7987	. 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 2178 {cab 2193 ⟨cop 3646 class class class wbr 4059 (class class class)co 5967 1_oc1o 6518 [cec 6641 Ncnpi 7420 <_N clti 7423 ~_Q ceq 7427 Qcnq 7428 <_Q cltq 7433 Pcnp 7439 1_Pc1p 7440 +_P cpp 7441 <_P cltp 7443 ~_R cer 7444 0_Rc0r 7446 <_R cltr 7451 <_ℝ cltrr 7964

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-coll 4175 ax-sep 4178 ax-nul 4186 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-setind 4603 ax-iinf 4654

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-ral 2491 df-rex 2492 df-reu 2493 df-rab 2495 df-v 2778 df-sbc 3006 df-csb 3102 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-nul 3469 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-int 3900 df-iun 3943 df-br 4060 df-opab 4122 df-mpt 4123 df-tr 4159 df-eprel 4354 df-id 4358 df-po 4361 df-iso 4362 df-iord 4431 df-on 4433 df-suc 4436 df-iom 4657 df-xp 4699 df-rel 4700 df-cnv 4701 df-co 4702 df-dm 4703 df-rn 4704 df-res 4705 df-ima 4706 df-iota 5251 df-fun 5292 df-fn 5293 df-f 5294 df-f1 5295 df-fo 5296 df-f1o 5297 df-fv 5298 df-ov 5970 df-oprab 5971 df-mpo 5972 df-1st 6249 df-2nd 6250 df-recs 6414 df-irdg 6479 df-1o 6525 df-2o 6526 df-oadd 6529 df-omul 6530 df-er 6643 df-ec 6645 df-qs 6649 df-ni 7452 df-pli 7453 df-mi 7454 df-lti 7455 df-plpq 7492 df-mpq 7493 df-enq 7495 df-nqqs 7496 df-plqqs 7497 df-mqqs 7498 df-1nqqs 7499 df-rq 7500 df-ltnqqs 7501 df-enq0 7572 df-nq0 7573 df-0nq0 7574 df-plq0 7575 df-mq0 7576 df-inp 7614 df-i1p 7615 df-iplp 7616 df-iltp 7618 df-enr 7874 df-nr 7875 df-ltr 7878 df-0r 7879 df-r 7970 df-lt 7973

This theorem is referenced by: ltrenn 8003 axcaucvglemres 8047

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