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Theorem ltprordil 7509
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 7425 . . . 4 <P ⊆ (P × P)
21brel 4638 . . 3 (𝐴<P 𝐵 → (𝐴P𝐵P))
3 ltdfpr 7426 . . . 4 ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ ∃𝑥Q (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵))))
43biimpd 143 . . 3 ((𝐴P𝐵P) → (𝐴<P 𝐵 → ∃𝑥Q (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵))))
52, 4mpcom 36 . 2 (𝐴<P 𝐵 → ∃𝑥Q (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))
6 simpll 519 . . . . . 6 (((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) ∧ 𝑦 ∈ (1st𝐴)) → 𝐴<P 𝐵)
7 simpr 109 . . . . . 6 (((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) ∧ 𝑦 ∈ (1st𝐴)) → 𝑦 ∈ (1st𝐴))
8 simprrl 529 . . . . . . 7 ((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) → 𝑥 ∈ (2nd𝐴))
98adantr 274 . . . . . 6 (((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) ∧ 𝑦 ∈ (1st𝐴)) → 𝑥 ∈ (2nd𝐴))
102simpld 111 . . . . . . . 8 (𝐴<P 𝐵𝐴P)
11 prop 7395 . . . . . . . 8 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
1210, 11syl 14 . . . . . . 7 (𝐴<P 𝐵 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
13 prltlu 7407 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (1st𝐴) ∧ 𝑥 ∈ (2nd𝐴)) → 𝑦 <Q 𝑥)
1412, 13syl3an1 1253 . . . . . 6 ((𝐴<P 𝐵𝑦 ∈ (1st𝐴) ∧ 𝑥 ∈ (2nd𝐴)) → 𝑦 <Q 𝑥)
156, 7, 9, 14syl3anc 1220 . . . . 5 (((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) ∧ 𝑦 ∈ (1st𝐴)) → 𝑦 <Q 𝑥)
16 simprrr 530 . . . . . . 7 ((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) → 𝑥 ∈ (1st𝐵))
1716adantr 274 . . . . . 6 (((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) ∧ 𝑦 ∈ (1st𝐴)) → 𝑥 ∈ (1st𝐵))
182simprd 113 . . . . . . . 8 (𝐴<P 𝐵𝐵P)
19 prop 7395 . . . . . . . 8 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2018, 19syl 14 . . . . . . 7 (𝐴<P 𝐵 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
21 prcdnql 7404 . . . . . . 7 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑥 ∈ (1st𝐵)) → (𝑦 <Q 𝑥𝑦 ∈ (1st𝐵)))
2220, 21sylan 281 . . . . . 6 ((𝐴<P 𝐵𝑥 ∈ (1st𝐵)) → (𝑦 <Q 𝑥𝑦 ∈ (1st𝐵)))
236, 17, 22syl2anc 409 . . . . 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 114 . . 3 ((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) → (𝑦 ∈ (1st𝐴) → 𝑦 ∈ (1st𝐵)))
2625ssrdv 3134 . 2 ((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) → (1st𝐴) ⊆ (1st𝐵))
275, 26rexlimddv 2579 1 (𝐴<P 𝐵 → (1st𝐴) ⊆ (1st𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 2128  wrex 2436  wss 3102  cop 3563   class class class wbr 3965  cfv 5170  1st c1st 6086  2nd c2nd 6087  Qcnq 7200   <Q cltq 7205  Pcnp 7211  <P cltp 7215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-oadd 6367  df-omul 6368  df-er 6480  df-ec 6482  df-qs 6486  df-ni 7224  df-mi 7226  df-lti 7227  df-enq 7267  df-nqqs 7268  df-ltnqqs 7273  df-inp 7386  df-iltp 7390
This theorem is referenced by:  ltexprlemrl  7530
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