ltprordil - Intuitionistic Logic Explorer

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Theorem ltprordil 7656

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 7572	. . . 4 ⊢ <_P ⊆ (P × P)
2	1	brel 4715	. . 3 ⊢ (𝐴<_P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
3		ltdfpr 7573	. . . 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 7542	. . . . . . . 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 7554	. . . . . . 7 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (1^st ‘𝐴) ∧ 𝑥 ∈ (2^nd ‘𝐴)) → 𝑦 <_Q 𝑥)
14	12, 13	syl3an1 1282	. . . . . 6 ⊢ ((𝐴<_P 𝐵 ∧ 𝑦 ∈ (1^st ‘𝐴) ∧ 𝑥 ∈ (2^nd ‘𝐴)) → 𝑦 <_Q 𝑥)
15	6, 7, 9, 14	syl3anc 1249	. . . . 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 7542	. . . . . . . 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 7551	. . . . . . 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 3189	. 2 ⊢ ((𝐴<_P 𝐵 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2^nd ‘𝐴) ∧ 𝑥 ∈ (1^st ‘𝐵)))) → (1^st ‘𝐴) ⊆ (1^st ‘𝐵))
27	5, 26	rexlimddv 2619	1 ⊢ (𝐴<_P 𝐵 → (1^st ‘𝐴) ⊆ (1^st ‘𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2167 ∃wrex 2476 ⊆ wss 3157 ⟨cop 3625 class class class wbr 4033 ‘cfv 5258 1^st c1st 6196 2^nd c2nd 6197 Qcnq 7347 <_Q cltq 7352 Pcnp 7358 <_P cltp 7362

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 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624

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 3451 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-eprel 4324 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-irdg 6428 df-oadd 6478 df-omul 6479 df-er 6592 df-ec 6594 df-qs 6598 df-ni 7371 df-mi 7373 df-lti 7374 df-enq 7414 df-nqqs 7415 df-ltnqqs 7420 df-inp 7533 df-iltp 7537

This theorem is referenced by: ltexprlemrl 7677

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