ltprordil - Intuitionistic Logic Explorer

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Theorem ltprordil 7684

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 7600	. . . 4 ⊢ <_P ⊆ (P × P)
2	1	brel 4725	. . 3 ⊢ (𝐴<_P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
3		ltdfpr 7601	. . . 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 7570	. . . . . . . 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 7582	. . . . . . 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 7570	. . . . . . . 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 7579	. . . . . . 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 3198	. 2 ⊢ ((𝐴<_P 𝐵 ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2^nd ‘𝐴) ∧ 𝑥 ∈ (1^st ‘𝐵)))) → (1^st ‘𝐴) ⊆ (1^st ‘𝐵))
27	5, 26	rexlimddv 2627	1 ⊢ (𝐴<_P 𝐵 → (1^st ‘𝐴) ⊆ (1^st ‘𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2175 ∃wrex 2484 ⊆ wss 3165 ⟨cop 3635 class class class wbr 4043 ‘cfv 5268 1^st c1st 6214 2^nd c2nd 6215 Qcnq 7375 <_Q cltq 7380 Pcnp 7386 <_P cltp 7390

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 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-coll 4158 ax-sep 4161 ax-nul 4169 ax-pow 4217 ax-pr 4252 ax-un 4478 ax-setind 4583 ax-iinf 4634

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-ral 2488 df-rex 2489 df-reu 2490 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-iun 3928 df-br 4044 df-opab 4105 df-mpt 4106 df-tr 4142 df-eprel 4334 df-id 4338 df-po 4341 df-iso 4342 df-iord 4411 df-on 4413 df-suc 4416 df-iom 4637 df-xp 4679 df-rel 4680 df-cnv 4681 df-co 4682 df-dm 4683 df-rn 4684 df-res 4685 df-ima 4686 df-iota 5229 df-fun 5270 df-fn 5271 df-f 5272 df-f1 5273 df-fo 5274 df-f1o 5275 df-fv 5276 df-ov 5937 df-oprab 5938 df-mpo 5939 df-1st 6216 df-2nd 6217 df-recs 6381 df-irdg 6446 df-oadd 6496 df-omul 6497 df-er 6610 df-ec 6612 df-qs 6616 df-ni 7399 df-mi 7401 df-lti 7402 df-enq 7442 df-nqqs 7443 df-ltnqqs 7448 df-inp 7561 df-iltp 7565

This theorem is referenced by: ltexprlemrl 7705

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