ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ltprordil GIF version

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 → (1st𝐴) ⊆ (1st𝐵))

Proof of Theorem ltprordil
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelpr 6603 . . . 4 <P ⊆ (P × P)
21brel 4392 . . 3 (𝐴<P 𝐵 → (𝐴P𝐵P))
3 ltdfpr 6604 . . . 4 ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ ∃𝑥Q (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵))))
43biimpd 132 . . 3 ((𝐴P𝐵P) → (𝐴<P 𝐵 → ∃𝑥Q (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵))))
52, 4mpcom 32 . 2 (𝐴<P 𝐵 → ∃𝑥Q (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))
6 simpll 481 . . . . . 6 (((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) ∧ 𝑦 ∈ (1st𝐴)) → 𝐴<P 𝐵)
7 simpr 103 . . . . . 6 (((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) ∧ 𝑦 ∈ (1st𝐴)) → 𝑦 ∈ (1st𝐴))
8 simprrl 491 . . . . . . 7 ((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) → 𝑥 ∈ (2nd𝐴))
98adantr 261 . . . . . 6 (((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) ∧ 𝑦 ∈ (1st𝐴)) → 𝑥 ∈ (2nd𝐴))
102simpld 105 . . . . . . . 8 (𝐴<P 𝐵𝐴P)
11 prop 6573 . . . . . . . 8 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
1210, 11syl 14 . . . . . . 7 (𝐴<P 𝐵 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
13 prltlu 6585 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (1st𝐴) ∧ 𝑥 ∈ (2nd𝐴)) → 𝑦 <Q 𝑥)
1412, 13syl3an1 1168 . . . . . 6 ((𝐴<P 𝐵𝑦 ∈ (1st𝐴) ∧ 𝑥 ∈ (2nd𝐴)) → 𝑦 <Q 𝑥)
156, 7, 9, 14syl3anc 1135 . . . . 5 (((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) ∧ 𝑦 ∈ (1st𝐴)) → 𝑦 <Q 𝑥)
16 simprrr 492 . . . . . . 7 ((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) → 𝑥 ∈ (1st𝐵))
1716adantr 261 . . . . . 6 (((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) ∧ 𝑦 ∈ (1st𝐴)) → 𝑥 ∈ (1st𝐵))
182simprd 107 . . . . . . . 8 (𝐴<P 𝐵𝐵P)
19 prop 6573 . . . . . . . 8 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2018, 19syl 14 . . . . . . 7 (𝐴<P 𝐵 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
21 prcdnql 6582 . . . . . . 7 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑥 ∈ (1st𝐵)) → (𝑦 <Q 𝑥𝑦 ∈ (1st𝐵)))
2220, 21sylan 267 . . . . . 6 ((𝐴<P 𝐵𝑥 ∈ (1st𝐵)) → (𝑦 <Q 𝑥𝑦 ∈ (1st𝐵)))
236, 17, 22syl2anc 391 . . . . 5 (((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) ∧ 𝑦 ∈ (1st𝐴)) → (𝑦 <Q 𝑥𝑦 ∈ (1st𝐵)))
2415, 23mpd 13 . . . 4 (((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) ∧ 𝑦 ∈ (1st𝐴)) → 𝑦 ∈ (1st𝐵))
2524ex 108 . . 3 ((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) → (𝑦 ∈ (1st𝐴) → 𝑦 ∈ (1st𝐵)))
2625ssrdv 2951 . 2 ((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) → (1st𝐴) ⊆ (1st𝐵))
275, 26rexlimddv 2437 1 (𝐴<P 𝐵 → (1st𝐴) ⊆ (1st𝐵))
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  1st c1st 5765  2nd 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
  Copyright terms: Public domain W3C validator