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Theorem ltpopr 6750
Description: Positive real 'less than' is a partial ordering. Remark ("< is transitive and irreflexive") preceding Proposition 11.2.3 of [HoTT], p. (varies). Lemma for ltsopr 6751. (Contributed by Jim Kingdon, 15-Dec-2019.)
Assertion
Ref Expression
ltpopr <P Po P

Proof of Theorem ltpopr
Dummy variables 𝑟 𝑞 𝑠 𝑡 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 6630 . . . . . . . 8 (𝑠P → ⟨(1st𝑠), (2nd𝑠)⟩ ∈ P)
2 prdisj 6647 . . . . . . . 8 ((⟨(1st𝑠), (2nd𝑠)⟩ ∈ P𝑞Q) → ¬ (𝑞 ∈ (1st𝑠) ∧ 𝑞 ∈ (2nd𝑠)))
31, 2sylan 271 . . . . . . 7 ((𝑠P𝑞Q) → ¬ (𝑞 ∈ (1st𝑠) ∧ 𝑞 ∈ (2nd𝑠)))
4 ancom 257 . . . . . . 7 ((𝑞 ∈ (1st𝑠) ∧ 𝑞 ∈ (2nd𝑠)) ↔ (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑠)))
53, 4sylnib 611 . . . . . 6 ((𝑠P𝑞Q) → ¬ (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑠)))
65nrexdv 2429 . . . . 5 (𝑠P → ¬ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑠)))
7 ltdfpr 6661 . . . . . 6 ((𝑠P𝑠P) → (𝑠<P 𝑠 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑠))))
87anidms 383 . . . . 5 (𝑠P → (𝑠<P 𝑠 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑠))))
96, 8mtbird 608 . . . 4 (𝑠P → ¬ 𝑠<P 𝑠)
109adantl 266 . . 3 ((⊤ ∧ 𝑠P) → ¬ 𝑠<P 𝑠)
11 ltdfpr 6661 . . . . . . . . . . 11 ((𝑠P𝑡P) → (𝑠<P 𝑡 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡))))
12113adant3 935 . . . . . . . . . 10 ((𝑠P𝑡P𝑢P) → (𝑠<P 𝑡 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡))))
13 ltdfpr 6661 . . . . . . . . . . 11 ((𝑡P𝑢P) → (𝑡<P 𝑢 ↔ ∃𝑟Q (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))))
14133adant1 933 . . . . . . . . . 10 ((𝑠P𝑡P𝑢P) → (𝑡<P 𝑢 ↔ ∃𝑟Q (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))))
1512, 14anbi12d 450 . . . . . . . . 9 ((𝑠P𝑡P𝑢P) → ((𝑠<P 𝑡𝑡<P 𝑢) ↔ (∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ ∃𝑟Q (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))))
16 reeanv 2496 . . . . . . . . 9 (∃𝑞Q𝑟Q ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) ↔ (∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ ∃𝑟Q (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))))
1715, 16syl6bbr 191 . . . . . . . 8 ((𝑠P𝑡P𝑢P) → ((𝑠<P 𝑡𝑡<P 𝑢) ↔ ∃𝑞Q𝑟Q ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))))
1817biimpa 284 . . . . . . 7 (((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) → ∃𝑞Q𝑟Q ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))))
19 simprll 497 . . . . . . . . . . 11 ((((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → 𝑞 ∈ (2nd𝑠))
20 prop 6630 . . . . . . . . . . . . . . . . . 18 (𝑡P → ⟨(1st𝑡), (2nd𝑡)⟩ ∈ P)
21 prltlu 6642 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝑡), (2nd𝑡)⟩ ∈ P𝑞 ∈ (1st𝑡) ∧ 𝑟 ∈ (2nd𝑡)) → 𝑞 <Q 𝑟)
2220, 21syl3an1 1179 . . . . . . . . . . . . . . . . 17 ((𝑡P𝑞 ∈ (1st𝑡) ∧ 𝑟 ∈ (2nd𝑡)) → 𝑞 <Q 𝑟)
23223adant3r 1143 . . . . . . . . . . . . . . . 16 ((𝑡P𝑞 ∈ (1st𝑡) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) → 𝑞 <Q 𝑟)
24233adant2l 1140 . . . . . . . . . . . . . . 15 ((𝑡P ∧ (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) → 𝑞 <Q 𝑟)
25243expb 1116 . . . . . . . . . . . . . 14 ((𝑡P ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → 𝑞 <Q 𝑟)
26253ad2antl2 1078 . . . . . . . . . . . . 13 (((𝑠P𝑡P𝑢P) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → 𝑞 <Q 𝑟)
2726adantlr 454 . . . . . . . . . . . 12 ((((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → 𝑞 <Q 𝑟)
28 prop 6630 . . . . . . . . . . . . . . . . 17 (𝑢P → ⟨(1st𝑢), (2nd𝑢)⟩ ∈ P)
29 prcdnql 6639 . . . . . . . . . . . . . . . . 17 ((⟨(1st𝑢), (2nd𝑢)⟩ ∈ P𝑟 ∈ (1st𝑢)) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑢)))
3028, 29sylan 271 . . . . . . . . . . . . . . . 16 ((𝑢P𝑟 ∈ (1st𝑢)) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑢)))
3130adantrl 455 . . . . . . . . . . . . . . 15 ((𝑢P ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑢)))
3231adantrl 455 . . . . . . . . . . . . . 14 ((𝑢P ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑢)))
33323ad2antl3 1079 . . . . . . . . . . . . 13 (((𝑠P𝑡P𝑢P) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑢)))
3433adantlr 454 . . . . . . . . . . . 12 ((((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑢)))
3527, 34mpd 13 . . . . . . . . . . 11 ((((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → 𝑞 ∈ (1st𝑢))
3619, 35jca 294 . . . . . . . . . 10 ((((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢)))
3736ex 112 . . . . . . . . 9 (((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) → (((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) → (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢))))
3837rexlimdvw 2453 . . . . . . . 8 (((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) → (∃𝑟Q ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) → (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢))))
3938reximdv 2437 . . . . . . 7 (((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) → (∃𝑞Q𝑟Q ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) → ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢))))
4018, 39mpd 13 . . . . . 6 (((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) → ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢)))
41 ltdfpr 6661 . . . . . . . . 9 ((𝑠P𝑢P) → (𝑠<P 𝑢 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢))))
42413adant2 934 . . . . . . . 8 ((𝑠P𝑡P𝑢P) → (𝑠<P 𝑢 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢))))
4342biimprd 151 . . . . . . 7 ((𝑠P𝑡P𝑢P) → (∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢)) → 𝑠<P 𝑢))
4443adantr 265 . . . . . 6 (((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) → (∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢)) → 𝑠<P 𝑢))
4540, 44mpd 13 . . . . 5 (((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) → 𝑠<P 𝑢)
4645ex 112 . . . 4 ((𝑠P𝑡P𝑢P) → ((𝑠<P 𝑡𝑡<P 𝑢) → 𝑠<P 𝑢))
4746adantl 266 . . 3 ((⊤ ∧ (𝑠P𝑡P𝑢P)) → ((𝑠<P 𝑡𝑡<P 𝑢) → 𝑠<P 𝑢))
4810, 47ispod 4068 . 2 (⊤ → <P Po P)
4948trud 1268 1 <P Po P
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  w3a 896  wtru 1260  wcel 1409  wrex 2324  cop 3405   class class class wbr 3791   Po wpo 4058  cfv 4929  1st c1st 5792  2nd c2nd 5793  Qcnq 6435   <Q cltq 6440  Pcnp 6446  <P cltp 6450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-mi 6461  df-lti 6462  df-enq 6502  df-nqqs 6503  df-ltnqqs 6508  df-inp 6621  df-iltp 6625
This theorem is referenced by:  ltsopr  6751
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