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Theorem ltpopr 7544
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 7545. (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 7424 . . . . . . . 8 (𝑠P → ⟨(1st𝑠), (2nd𝑠)⟩ ∈ P)
2 prdisj 7441 . . . . . . . 8 ((⟨(1st𝑠), (2nd𝑠)⟩ ∈ P𝑞Q) → ¬ (𝑞 ∈ (1st𝑠) ∧ 𝑞 ∈ (2nd𝑠)))
31, 2sylan 281 . . . . . . 7 ((𝑠P𝑞Q) → ¬ (𝑞 ∈ (1st𝑠) ∧ 𝑞 ∈ (2nd𝑠)))
4 ancom 264 . . . . . . 7 ((𝑞 ∈ (1st𝑠) ∧ 𝑞 ∈ (2nd𝑠)) ↔ (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑠)))
53, 4sylnib 671 . . . . . 6 ((𝑠P𝑞Q) → ¬ (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑠)))
65nrexdv 2563 . . . . 5 (𝑠P → ¬ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑠)))
7 ltdfpr 7455 . . . . . 6 ((𝑠P𝑠P) → (𝑠<P 𝑠 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑠))))
87anidms 395 . . . . 5 (𝑠P → (𝑠<P 𝑠 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑠))))
96, 8mtbird 668 . . . 4 (𝑠P → ¬ 𝑠<P 𝑠)
109adantl 275 . . 3 ((⊤ ∧ 𝑠P) → ¬ 𝑠<P 𝑠)
11 ltdfpr 7455 . . . . . . . . . . 11 ((𝑠P𝑡P) → (𝑠<P 𝑡 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡))))
12113adant3 1012 . . . . . . . . . 10 ((𝑠P𝑡P𝑢P) → (𝑠<P 𝑡 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡))))
13 ltdfpr 7455 . . . . . . . . . . 11 ((𝑡P𝑢P) → (𝑡<P 𝑢 ↔ ∃𝑟Q (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))))
14133adant1 1010 . . . . . . . . . 10 ((𝑠P𝑡P𝑢P) → (𝑡<P 𝑢 ↔ ∃𝑟Q (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))))
1512, 14anbi12d 470 . . . . . . . . 9 ((𝑠P𝑡P𝑢P) → ((𝑠<P 𝑡𝑡<P 𝑢) ↔ (∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ ∃𝑟Q (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))))
16 reeanv 2639 . . . . . . . . 9 (∃𝑞Q𝑟Q ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) ↔ (∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ ∃𝑟Q (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))))
1715, 16bitr4di 197 . . . . . . . 8 ((𝑠P𝑡P𝑢P) → ((𝑠<P 𝑡𝑡<P 𝑢) ↔ ∃𝑞Q𝑟Q ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))))
1817biimpa 294 . . . . . . 7 (((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) → ∃𝑞Q𝑟Q ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))))
19 simprll 532 . . . . . . . . . . 11 ((((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → 𝑞 ∈ (2nd𝑠))
20 prop 7424 . . . . . . . . . . . . . . . . . 18 (𝑡P → ⟨(1st𝑡), (2nd𝑡)⟩ ∈ P)
21 prltlu 7436 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝑡), (2nd𝑡)⟩ ∈ P𝑞 ∈ (1st𝑡) ∧ 𝑟 ∈ (2nd𝑡)) → 𝑞 <Q 𝑟)
2220, 21syl3an1 1266 . . . . . . . . . . . . . . . . 17 ((𝑡P𝑞 ∈ (1st𝑡) ∧ 𝑟 ∈ (2nd𝑡)) → 𝑞 <Q 𝑟)
23223adant3r 1230 . . . . . . . . . . . . . . . 16 ((𝑡P𝑞 ∈ (1st𝑡) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) → 𝑞 <Q 𝑟)
24233adant2l 1227 . . . . . . . . . . . . . . 15 ((𝑡P ∧ (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) → 𝑞 <Q 𝑟)
25243expb 1199 . . . . . . . . . . . . . 14 ((𝑡P ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → 𝑞 <Q 𝑟)
26253ad2antl2 1155 . . . . . . . . . . . . 13 (((𝑠P𝑡P𝑢P) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → 𝑞 <Q 𝑟)
2726adantlr 474 . . . . . . . . . . . 12 ((((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → 𝑞 <Q 𝑟)
28 prop 7424 . . . . . . . . . . . . . . . . 17 (𝑢P → ⟨(1st𝑢), (2nd𝑢)⟩ ∈ P)
29 prcdnql 7433 . . . . . . . . . . . . . . . . 17 ((⟨(1st𝑢), (2nd𝑢)⟩ ∈ P𝑟 ∈ (1st𝑢)) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑢)))
3028, 29sylan 281 . . . . . . . . . . . . . . . 16 ((𝑢P𝑟 ∈ (1st𝑢)) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑢)))
3130adantrl 475 . . . . . . . . . . . . . . 15 ((𝑢P ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑢)))
3231adantrl 475 . . . . . . . . . . . . . 14 ((𝑢P ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑢)))
33323ad2antl3 1156 . . . . . . . . . . . . 13 (((𝑠P𝑡P𝑢P) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑢)))
3433adantlr 474 . . . . . . . . . . . 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 304 . . . . . . . . . 10 ((((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢)))
3736ex 114 . . . . . . . . 9 (((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) → (((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) → (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢))))
3837rexlimdvw 2591 . . . . . . . 8 (((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) → (∃𝑟Q ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) → (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢))))
3938reximdv 2571 . . . . . . 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 7455 . . . . . . . . 9 ((𝑠P𝑢P) → (𝑠<P 𝑢 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢))))
42413adant2 1011 . . . . . . . 8 ((𝑠P𝑡P𝑢P) → (𝑠<P 𝑢 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢))))
4342biimprd 157 . . . . . . 7 ((𝑠P𝑡P𝑢P) → (∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢)) → 𝑠<P 𝑢))
4443adantr 274 . . . . . 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 114 . . . 4 ((𝑠P𝑡P𝑢P) → ((𝑠<P 𝑡𝑡<P 𝑢) → 𝑠<P 𝑢))
4746adantl 275 . . 3 ((⊤ ∧ (𝑠P𝑡P𝑢P)) → ((𝑠<P 𝑡𝑡<P 𝑢) → 𝑠<P 𝑢))
4810, 47ispod 4287 . 2 (⊤ → <P Po P)
4948mptru 1357 1 <P Po P
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  w3a 973  wtru 1349  wcel 2141  wrex 2449  cop 3584   class class class wbr 3987   Po wpo 4277  cfv 5196  1st c1st 6114  2nd c2nd 6115  Qcnq 7229   <Q cltq 7234  Pcnp 7240  <P cltp 7244
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-eprel 4272  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-irdg 6346  df-oadd 6396  df-omul 6397  df-er 6509  df-ec 6511  df-qs 6515  df-ni 7253  df-mi 7255  df-lti 7256  df-enq 7296  df-nqqs 7297  df-ltnqqs 7302  df-inp 7415  df-iltp 7419
This theorem is referenced by:  ltsopr  7545
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