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Theorem ltpopr 7608
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 7609. (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 7488 . . . . . . . 8 (𝑠P → ⟨(1st𝑠), (2nd𝑠)⟩ ∈ P)
2 prdisj 7505 . . . . . . . 8 ((⟨(1st𝑠), (2nd𝑠)⟩ ∈ P𝑞Q) → ¬ (𝑞 ∈ (1st𝑠) ∧ 𝑞 ∈ (2nd𝑠)))
31, 2sylan 283 . . . . . . 7 ((𝑠P𝑞Q) → ¬ (𝑞 ∈ (1st𝑠) ∧ 𝑞 ∈ (2nd𝑠)))
4 ancom 266 . . . . . . 7 ((𝑞 ∈ (1st𝑠) ∧ 𝑞 ∈ (2nd𝑠)) ↔ (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑠)))
53, 4sylnib 677 . . . . . 6 ((𝑠P𝑞Q) → ¬ (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑠)))
65nrexdv 2580 . . . . 5 (𝑠P → ¬ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑠)))
7 ltdfpr 7519 . . . . . 6 ((𝑠P𝑠P) → (𝑠<P 𝑠 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑠))))
87anidms 397 . . . . 5 (𝑠P → (𝑠<P 𝑠 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑠))))
96, 8mtbird 674 . . . 4 (𝑠P → ¬ 𝑠<P 𝑠)
109adantl 277 . . 3 ((⊤ ∧ 𝑠P) → ¬ 𝑠<P 𝑠)
11 ltdfpr 7519 . . . . . . . . . . 11 ((𝑠P𝑡P) → (𝑠<P 𝑡 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡))))
12113adant3 1018 . . . . . . . . . 10 ((𝑠P𝑡P𝑢P) → (𝑠<P 𝑡 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡))))
13 ltdfpr 7519 . . . . . . . . . . 11 ((𝑡P𝑢P) → (𝑡<P 𝑢 ↔ ∃𝑟Q (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))))
14133adant1 1016 . . . . . . . . . 10 ((𝑠P𝑡P𝑢P) → (𝑡<P 𝑢 ↔ ∃𝑟Q (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))))
1512, 14anbi12d 473 . . . . . . . . 9 ((𝑠P𝑡P𝑢P) → ((𝑠<P 𝑡𝑡<P 𝑢) ↔ (∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ ∃𝑟Q (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))))
16 reeanv 2657 . . . . . . . . 9 (∃𝑞Q𝑟Q ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) ↔ (∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ ∃𝑟Q (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))))
1715, 16bitr4di 198 . . . . . . . 8 ((𝑠P𝑡P𝑢P) → ((𝑠<P 𝑡𝑡<P 𝑢) ↔ ∃𝑞Q𝑟Q ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))))
1817biimpa 296 . . . . . . 7 (((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) → ∃𝑞Q𝑟Q ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))))
19 simprll 537 . . . . . . . . . . 11 ((((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → 𝑞 ∈ (2nd𝑠))
20 prop 7488 . . . . . . . . . . . . . . . . . 18 (𝑡P → ⟨(1st𝑡), (2nd𝑡)⟩ ∈ P)
21 prltlu 7500 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝑡), (2nd𝑡)⟩ ∈ P𝑞 ∈ (1st𝑡) ∧ 𝑟 ∈ (2nd𝑡)) → 𝑞 <Q 𝑟)
2220, 21syl3an1 1281 . . . . . . . . . . . . . . . . 17 ((𝑡P𝑞 ∈ (1st𝑡) ∧ 𝑟 ∈ (2nd𝑡)) → 𝑞 <Q 𝑟)
23223adant3r 1236 . . . . . . . . . . . . . . . 16 ((𝑡P𝑞 ∈ (1st𝑡) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) → 𝑞 <Q 𝑟)
24233adant2l 1233 . . . . . . . . . . . . . . 15 ((𝑡P ∧ (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) → 𝑞 <Q 𝑟)
25243expb 1205 . . . . . . . . . . . . . 14 ((𝑡P ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → 𝑞 <Q 𝑟)
26253ad2antl2 1161 . . . . . . . . . . . . 13 (((𝑠P𝑡P𝑢P) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → 𝑞 <Q 𝑟)
2726adantlr 477 . . . . . . . . . . . 12 ((((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → 𝑞 <Q 𝑟)
28 prop 7488 . . . . . . . . . . . . . . . . 17 (𝑢P → ⟨(1st𝑢), (2nd𝑢)⟩ ∈ P)
29 prcdnql 7497 . . . . . . . . . . . . . . . . 17 ((⟨(1st𝑢), (2nd𝑢)⟩ ∈ P𝑟 ∈ (1st𝑢)) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑢)))
3028, 29sylan 283 . . . . . . . . . . . . . . . 16 ((𝑢P𝑟 ∈ (1st𝑢)) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑢)))
3130adantrl 478 . . . . . . . . . . . . . . 15 ((𝑢P ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑢)))
3231adantrl 478 . . . . . . . . . . . . . 14 ((𝑢P ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑢)))
33323ad2antl3 1162 . . . . . . . . . . . . 13 (((𝑠P𝑡P𝑢P) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑢)))
3433adantlr 477 . . . . . . . . . . . 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 306 . . . . . . . . . 10 ((((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢)))
3736ex 115 . . . . . . . . 9 (((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) → (((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) → (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢))))
3837rexlimdvw 2608 . . . . . . . 8 (((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) → (∃𝑟Q ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) → (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢))))
3938reximdv 2588 . . . . . . 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 7519 . . . . . . . . 9 ((𝑠P𝑢P) → (𝑠<P 𝑢 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢))))
42413adant2 1017 . . . . . . . 8 ((𝑠P𝑡P𝑢P) → (𝑠<P 𝑢 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢))))
4342biimprd 158 . . . . . . 7 ((𝑠P𝑡P𝑢P) → (∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢)) → 𝑠<P 𝑢))
4443adantr 276 . . . . . 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 115 . . . 4 ((𝑠P𝑡P𝑢P) → ((𝑠<P 𝑡𝑡<P 𝑢) → 𝑠<P 𝑢))
4746adantl 277 . . 3 ((⊤ ∧ (𝑠P𝑡P𝑢P)) → ((𝑠<P 𝑡𝑡<P 𝑢) → 𝑠<P 𝑢))
4810, 47ispod 4316 . 2 (⊤ → <P Po P)
4948mptru 1372 1 <P Po P
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  w3a 979  wtru 1364  wcel 2158  wrex 2466  cop 3607   class class class wbr 4015   Po wpo 4306  cfv 5228  1st c1st 6153  2nd c2nd 6154  Qcnq 7293   <Q cltq 7298  Pcnp 7304  <P cltp 7308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-oadd 6435  df-omul 6436  df-er 6549  df-ec 6551  df-qs 6555  df-ni 7317  df-mi 7319  df-lti 7320  df-enq 7360  df-nqqs 7361  df-ltnqqs 7366  df-inp 7479  df-iltp 7483
This theorem is referenced by:  ltsopr  7609
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