ltprordil - Intuitionistic Logic Explorer

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Theorem ltprordil 6687

Description: If a positive real is less than a second positive real, its lower cut is a subset of the second's lower cut. (Contributed by Jim Kingdon, 23-Dec-2019.)

**Assertion**
Ref	Expression
ltprordil	⊢ (𝐴<_P 𝐵 → (1^st ‘𝐴) ⊆ (1^st ‘𝐵))

Proof of Theorem ltprordil Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelpr 6603	. . . 4 ⊢ <_P ⊆ (P × P)
2	1	brel 4392	. . 3 ⊢ (𝐴<_P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
3		ltdfpr 6604	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<_P 𝐵 ↔ ∃𝑥 ∈ Q (𝑥 ∈ (2^nd ‘𝐴) ∧ 𝑥 ∈ (1^st ‘𝐵))))
4	3	biimpd 132	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<_P 𝐵 → ∃𝑥 ∈ Q (𝑥 ∈ (2^nd ‘𝐴) ∧ 𝑥 ∈ (1^st ‘𝐵))))
5	2, 4	mpcom 32	. 2 ⊢ (𝐴<_P 𝐵 → ∃𝑥 ∈ Q (𝑥 ∈ (2^nd ‘𝐴) ∧ 𝑥 ∈ (1^st ‘𝐵)))
6		simpll 481	. . . . . 6 ⊢ (((𝐴<_P 𝐵 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2^nd ‘𝐴) ∧ 𝑥 ∈ (1^st ‘𝐵)))) ∧ 𝑦 ∈ (1^st ‘𝐴)) → 𝐴<_P 𝐵)
7		simpr 103	. . . . . 6 ⊢ (((𝐴<_P 𝐵 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2^nd ‘𝐴) ∧ 𝑥 ∈ (1^st ‘𝐵)))) ∧ 𝑦 ∈ (1^st ‘𝐴)) → 𝑦 ∈ (1^st ‘𝐴))
8		simprrl 491	. . . . . . 7 ⊢ ((𝐴<_P 𝐵 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2^nd ‘𝐴) ∧ 𝑥 ∈ (1^st ‘𝐵)))) → 𝑥 ∈ (2^nd ‘𝐴))
9	8	adantr 261	. . . . . 6 ⊢ (((𝐴<_P 𝐵 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2^nd ‘𝐴) ∧ 𝑥 ∈ (1^st ‘𝐵)))) ∧ 𝑦 ∈ (1^st ‘𝐴)) → 𝑥 ∈ (2^nd ‘𝐴))
10	2	simpld 105	. . . . . . . 8 ⊢ (𝐴<_P 𝐵 → 𝐴 ∈ P)
11		prop 6573	. . . . . . . 8 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
12	10, 11	syl 14	. . . . . . 7 ⊢ (𝐴<_P 𝐵 → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
13		prltlu 6585	. . . . . . 7 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (1^st ‘𝐴) ∧ 𝑥 ∈ (2^nd ‘𝐴)) → 𝑦 <_Q 𝑥)
14	12, 13	syl3an1 1168	. . . . . 6 ⊢ ((𝐴<_P 𝐵 ∧ 𝑦 ∈ (1^st ‘𝐴) ∧ 𝑥 ∈ (2^nd ‘𝐴)) → 𝑦 <_Q 𝑥)
15	6, 7, 9, 14	syl3anc 1135	. . . . 5 ⊢ (((𝐴<_P 𝐵 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2^nd ‘𝐴) ∧ 𝑥 ∈ (1^st ‘𝐵)))) ∧ 𝑦 ∈ (1^st ‘𝐴)) → 𝑦 <_Q 𝑥)
16		simprrr 492	. . . . . . 7 ⊢ ((𝐴<_P 𝐵 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2^nd ‘𝐴) ∧ 𝑥 ∈ (1^st ‘𝐵)))) → 𝑥 ∈ (1^st ‘𝐵))
17	16	adantr 261	. . . . . 6 ⊢ (((𝐴<_P 𝐵 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2^nd ‘𝐴) ∧ 𝑥 ∈ (1^st ‘𝐵)))) ∧ 𝑦 ∈ (1^st ‘𝐴)) → 𝑥 ∈ (1^st ‘𝐵))
18	2	simprd 107	. . . . . . . 8 ⊢ (𝐴<_P 𝐵 → 𝐵 ∈ P)
19		prop 6573	. . . . . . . 8 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
20	18, 19	syl 14	. . . . . . 7 ⊢ (𝐴<_P 𝐵 → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
21		prcdnql 6582	. . . . . . 7 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝑥 ∈ (1^st ‘𝐵)) → (𝑦 <_Q 𝑥 → 𝑦 ∈ (1^st ‘𝐵)))
22	20, 21	sylan 267	. . . . . 6 ⊢ ((𝐴<_P 𝐵 ∧ 𝑥 ∈ (1^st ‘𝐵)) → (𝑦 <_Q 𝑥 → 𝑦 ∈ (1^st ‘𝐵)))
23	6, 17, 22	syl2anc 391	. . . . 5 ⊢ (((𝐴<_P 𝐵 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2^nd ‘𝐴) ∧ 𝑥 ∈ (1^st ‘𝐵)))) ∧ 𝑦 ∈ (1^st ‘𝐴)) → (𝑦 <_Q 𝑥 → 𝑦 ∈ (1^st ‘𝐵)))
24	15, 23	mpd 13	. . . 4 ⊢ (((𝐴<_P 𝐵 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2^nd ‘𝐴) ∧ 𝑥 ∈ (1^st ‘𝐵)))) ∧ 𝑦 ∈ (1^st ‘𝐴)) → 𝑦 ∈ (1^st ‘𝐵))
25	24	ex 108	. . 3 ⊢ ((𝐴<_P 𝐵 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2^nd ‘𝐴) ∧ 𝑥 ∈ (1^st ‘𝐵)))) → (𝑦 ∈ (1^st ‘𝐴) → 𝑦 ∈ (1^st ‘𝐵)))
26	25	ssrdv 2951	. 2 ⊢ ((𝐴<_P 𝐵 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2^nd ‘𝐴) ∧ 𝑥 ∈ (1^st ‘𝐵)))) → (1^st ‘𝐴) ⊆ (1^st ‘𝐵))
27	5, 26	rexlimddv 2437	1 ⊢ (𝐴<_P 𝐵 → (1^st ‘𝐴) ⊆ (1^st ‘𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1393 ∃wrex 2307 ⊆ wss 2917 ⟨cop 3378 class class class wbr 3764 ‘cfv 4902 1^st c1st 5765 2^nd c2nd 5766 Qcnq 6378 <_Q cltq 6383 Pcnp 6389 <_P cltp 6393

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311

This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-mi 6404 df-lti 6405 df-enq 6445 df-nqqs 6446 df-ltnqqs 6451 df-inp 6564 df-iltp 6568

This theorem is referenced by: ltexprlemrl 6708

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