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Theorem ltprordil 6630
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 6546 . . . 4 <P ⊆ (P × P)
21brel 4353 . . 3 (𝐴<P 𝐵 → (𝐴P𝐵P))
3 ltdfpr 6547 . . . 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 6516 . . . . . . . 8 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
1210, 11syl 14 . . . . . . 7 (𝐴<P 𝐵 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
13 prltlu 6528 . . . . . . 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 6516 . . . . . . . 8 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2018, 19syl 14 . . . . . . 7 (𝐴<P 𝐵 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
21 prcdnql 6525 . . . . . . 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 2948 . 2 ((𝐴<P 𝐵 ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑥 ∈ (1st𝐵)))) → (1st𝐴) ⊆ (1st𝐵))
275, 26rexlimddv 2434 1 (𝐴<P 𝐵 → (1st𝐴) ⊆ (1st𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wcel 1393  wrex 2304  wss 2914  cop 3375   class class class wbr 3760  cfv 4863  1st c1st 5726  2nd c2nd 5727  Qcnq 6321   <Q cltq 6326  Pcnp 6332  <P cltp 6336
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 3868  ax-sep 3871  ax-nul 3879  ax-pow 3923  ax-pr 3940  ax-un 4141  ax-setind 4230  ax-iinf 4272
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 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3577  df-int 3612  df-iun 3655  df-br 3761  df-opab 3815  df-mpt 3816  df-tr 3851  df-eprel 4022  df-id 4026  df-po 4029  df-iso 4030  df-iord 4074  df-on 4076  df-suc 4079  df-iom 4275  df-xp 4312  df-rel 4313  df-cnv 4314  df-co 4315  df-dm 4316  df-rn 4317  df-res 4318  df-ima 4319  df-iota 4828  df-fun 4865  df-fn 4866  df-f 4867  df-f1 4868  df-fo 4869  df-f1o 4870  df-fv 4871  df-ov 5476  df-oprab 5477  df-mpt2 5478  df-1st 5728  df-2nd 5729  df-recs 5881  df-irdg 5918  df-oadd 5966  df-omul 5967  df-er 6065  df-ec 6067  df-qs 6071  df-ni 6345  df-mi 6347  df-lti 6348  df-enq 6388  df-nqqs 6389  df-ltnqqs 6394  df-inp 6507  df-iltp 6511
This theorem is referenced by:  ltexprlemrl  6651
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