ltprordil - Intuitionistic Logic Explorer

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Theorem ltprordil 7797

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 7713	. . . 4 ⊢ <_P ⊆ (P × P)
2	1	brel 4774	. . 3 ⊢ (𝐴<_P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
3		ltdfpr 7714	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<_P 𝐵 ↔ ∃𝑥 ∈ Q (𝑥 ∈ (2^nd ‘𝐴) ∧ 𝑥 ∈ (1^st ‘𝐵))))
4	3	biimpd 144	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<_P 𝐵 → ∃𝑥 ∈ Q (𝑥 ∈ (2^nd ‘𝐴) ∧ 𝑥 ∈ (1^st ‘𝐵))))
5	2, 4	mpcom 36	. 2 ⊢ (𝐴<_P 𝐵 → ∃𝑥 ∈ Q (𝑥 ∈ (2^nd ‘𝐴) ∧ 𝑥 ∈ (1^st ‘𝐵)))
6		simpll 527	. . . . . 6 ⊢ (((𝐴<_P 𝐵 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2^nd ‘𝐴) ∧ 𝑥 ∈ (1^st ‘𝐵)))) ∧ 𝑦 ∈ (1^st ‘𝐴)) → 𝐴<_P 𝐵)
7		simpr 110	. . . . . 6 ⊢ (((𝐴<_P 𝐵 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2^nd ‘𝐴) ∧ 𝑥 ∈ (1^st ‘𝐵)))) ∧ 𝑦 ∈ (1^st ‘𝐴)) → 𝑦 ∈ (1^st ‘𝐴))
8		simprrl 539	. . . . . . 7 ⊢ ((𝐴<_P 𝐵 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2^nd ‘𝐴) ∧ 𝑥 ∈ (1^st ‘𝐵)))) → 𝑥 ∈ (2^nd ‘𝐴))
9	8	adantr 276	. . . . . 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 112	. . . . . . . 8 ⊢ (𝐴<_P 𝐵 → 𝐴 ∈ P)
11		prop 7683	. . . . . . . 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 7695	. . . . . . 7 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (1^st ‘𝐴) ∧ 𝑥 ∈ (2^nd ‘𝐴)) → 𝑦 <_Q 𝑥)
14	12, 13	syl3an1 1304	. . . . . 6 ⊢ ((𝐴<_P 𝐵 ∧ 𝑦 ∈ (1^st ‘𝐴) ∧ 𝑥 ∈ (2^nd ‘𝐴)) → 𝑦 <_Q 𝑥)
15	6, 7, 9, 14	syl3anc 1271	. . . . 5 ⊢ (((𝐴<_P 𝐵 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2^nd ‘𝐴) ∧ 𝑥 ∈ (1^st ‘𝐵)))) ∧ 𝑦 ∈ (1^st ‘𝐴)) → 𝑦 <_Q 𝑥)
16		simprrr 540	. . . . . . 7 ⊢ ((𝐴<_P 𝐵 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2^nd ‘𝐴) ∧ 𝑥 ∈ (1^st ‘𝐵)))) → 𝑥 ∈ (1^st ‘𝐵))
17	16	adantr 276	. . . . . 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 114	. . . . . . . 8 ⊢ (𝐴<_P 𝐵 → 𝐵 ∈ P)
19		prop 7683	. . . . . . . 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 7692	. . . . . . 7 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝑥 ∈ (1^st ‘𝐵)) → (𝑦 <_Q 𝑥 → 𝑦 ∈ (1^st ‘𝐵)))
22	20, 21	sylan 283	. . . . . 6 ⊢ ((𝐴<_P 𝐵 ∧ 𝑥 ∈ (1^st ‘𝐵)) → (𝑦 <_Q 𝑥 → 𝑦 ∈ (1^st ‘𝐵)))
23	6, 17, 22	syl2anc 411	. . . . 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 115	. . 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 3231	. 2 ⊢ ((𝐴<_P 𝐵 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2^nd ‘𝐴) ∧ 𝑥 ∈ (1^st ‘𝐵)))) → (1^st ‘𝐴) ⊆ (1^st ‘𝐵))
27	5, 26	rexlimddv 2653	1 ⊢ (𝐴<_P 𝐵 → (1^st ‘𝐴) ⊆ (1^st ‘𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2200 ∃wrex 2509 ⊆ wss 3198 ⟨cop 3670 class class class wbr 4084 ‘cfv 5322 1^st c1st 6294 2^nd c2nd 6295 Qcnq 7488 <_Q cltq 7493 Pcnp 7499 <_P cltp 7503

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4200 ax-sep 4203 ax-nul 4211 ax-pow 4260 ax-pr 4295 ax-un 4526 ax-setind 4631 ax-iinf 4682

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3890 df-int 3925 df-iun 3968 df-br 4085 df-opab 4147 df-mpt 4148 df-tr 4184 df-eprel 4382 df-id 4386 df-po 4389 df-iso 4390 df-iord 4459 df-on 4461 df-suc 4464 df-iom 4685 df-xp 4727 df-rel 4728 df-cnv 4729 df-co 4730 df-dm 4731 df-rn 4732 df-res 4733 df-ima 4734 df-iota 5282 df-fun 5324 df-fn 5325 df-f 5326 df-f1 5327 df-fo 5328 df-f1o 5329 df-fv 5330 df-ov 6014 df-oprab 6015 df-mpo 6016 df-1st 6296 df-2nd 6297 df-recs 6464 df-irdg 6529 df-oadd 6579 df-omul 6580 df-er 6695 df-ec 6697 df-qs 6701 df-ni 7512 df-mi 7514 df-lti 7515 df-enq 7555 df-nqqs 7556 df-ltnqqs 7561 df-inp 7674 df-iltp 7678

This theorem is referenced by: ltexprlemrl 7818

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