ltprordil - Intuitionistic Logic Explorer

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Theorem ltprordil 7852

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 7768	. . . 4 ⊢ <_P ⊆ (P × P)
2	1	brel 4784	. . 3 ⊢ (𝐴<_P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
3		ltdfpr 7769	. . . 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 541	. . . . . . 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 7738	. . . . . . . 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 7750	. . . . . . 7 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (1^st ‘𝐴) ∧ 𝑥 ∈ (2^nd ‘𝐴)) → 𝑦 <_Q 𝑥)
14	12, 13	syl3an1 1307	. . . . . 6 ⊢ ((𝐴<_P 𝐵 ∧ 𝑦 ∈ (1^st ‘𝐴) ∧ 𝑥 ∈ (2^nd ‘𝐴)) → 𝑦 <_Q 𝑥)
15	6, 7, 9, 14	syl3anc 1274	. . . . 5 ⊢ (((𝐴<_P 𝐵 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2^nd ‘𝐴) ∧ 𝑥 ∈ (1^st ‘𝐵)))) ∧ 𝑦 ∈ (1^st ‘𝐴)) → 𝑦 <_Q 𝑥)
16		simprrr 542	. . . . . . 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 7738	. . . . . . . 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 7747	. . . . . . 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 3234	. 2 ⊢ ((𝐴<_P 𝐵 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2^nd ‘𝐴) ∧ 𝑥 ∈ (1^st ‘𝐵)))) → (1^st ‘𝐴) ⊆ (1^st ‘𝐵))
27	5, 26	rexlimddv 2656	1 ⊢ (𝐴<_P 𝐵 → (1^st ‘𝐴) ⊆ (1^st ‘𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2202 ∃wrex 2512 ⊆ wss 3201 ⟨cop 3676 class class class wbr 4093 ‘cfv 5333 1^st c1st 6310 2^nd c2nd 6311 Qcnq 7543 <_Q cltq 7548 Pcnp 7554 <_P cltp 7558

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-ral 2516 df-rex 2517 df-reu 2518 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-eprel 4392 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-irdg 6579 df-oadd 6629 df-omul 6630 df-er 6745 df-ec 6747 df-qs 6751 df-ni 7567 df-mi 7569 df-lti 7570 df-enq 7610 df-nqqs 7611 df-ltnqqs 7616 df-inp 7729 df-iltp 7733

This theorem is referenced by: ltexprlemrl 7873

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