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Theorem xrlttri 11916
Description: Ordering on the extended reals satisfies strict trichotomy. New proofs should generally use this instead of ax-pre-lttri 9954 or axlttri 10053. (Contributed by NM, 14-Oct-2005.)
Assertion
Ref Expression
xrlttri ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴)))

Proof of Theorem xrlttri
StepHypRef Expression
1 xrltnr 11897 . . . . . . . 8 (𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴)
21adantr 481 . . . . . . 7 ((𝐴 ∈ ℝ*𝐴 = 𝐵) → ¬ 𝐴 < 𝐴)
3 breq2 4617 . . . . . . . 8 (𝐴 = 𝐵 → (𝐴 < 𝐴𝐴 < 𝐵))
43adantl 482 . . . . . . 7 ((𝐴 ∈ ℝ*𝐴 = 𝐵) → (𝐴 < 𝐴𝐴 < 𝐵))
52, 4mtbid 314 . . . . . 6 ((𝐴 ∈ ℝ*𝐴 = 𝐵) → ¬ 𝐴 < 𝐵)
65ex 450 . . . . 5 (𝐴 ∈ ℝ* → (𝐴 = 𝐵 → ¬ 𝐴 < 𝐵))
76adantr 481 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 = 𝐵 → ¬ 𝐴 < 𝐵))
8 xrltnsym 11914 . . . . 5 ((𝐵 ∈ ℝ*𝐴 ∈ ℝ*) → (𝐵 < 𝐴 → ¬ 𝐴 < 𝐵))
98ancoms 469 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐵 < 𝐴 → ¬ 𝐴 < 𝐵))
107, 9jaod 395 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴 = 𝐵𝐵 < 𝐴) → ¬ 𝐴 < 𝐵))
11 elxr 11894 . . . 4 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
12 elxr 11894 . . . 4 (𝐵 ∈ ℝ* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
13 axlttri 10053 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴)))
1413biimprd 238 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ (𝐴 = 𝐵𝐵 < 𝐴) → 𝐴 < 𝐵))
1514con1d 139 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
16 ltpnf 11898 . . . . . . . . 9 (𝐴 ∈ ℝ → 𝐴 < +∞)
1716adantr 481 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐴 < +∞)
18 breq2 4617 . . . . . . . . 9 (𝐵 = +∞ → (𝐴 < 𝐵𝐴 < +∞))
1918adantl 482 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 < 𝐵𝐴 < +∞))
2017, 19mpbird 247 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
2120pm2.24d 147 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
22 mnflt 11901 . . . . . . . . . 10 (𝐴 ∈ ℝ → -∞ < 𝐴)
2322adantr 481 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → -∞ < 𝐴)
24 breq1 4616 . . . . . . . . . 10 (𝐵 = -∞ → (𝐵 < 𝐴 ↔ -∞ < 𝐴))
2524adantl 482 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐵 < 𝐴 ↔ -∞ < 𝐴))
2623, 25mpbird 247 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → 𝐵 < 𝐴)
2726olcd 408 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 = 𝐵𝐵 < 𝐴))
2827a1d 25 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
2915, 21, 283jaodan 1391 . . . . 5 ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
30 ltpnf 11898 . . . . . . . . . 10 (𝐵 ∈ ℝ → 𝐵 < +∞)
3130adantl 482 . . . . . . . . 9 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → 𝐵 < +∞)
32 breq2 4617 . . . . . . . . . 10 (𝐴 = +∞ → (𝐵 < 𝐴𝐵 < +∞))
3332adantr 481 . . . . . . . . 9 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴𝐵 < +∞))
3431, 33mpbird 247 . . . . . . . 8 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → 𝐵 < 𝐴)
3534olcd 408 . . . . . . 7 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵𝐵 < 𝐴))
3635a1d 25 . . . . . 6 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
37 eqtr3 2642 . . . . . . . 8 ((𝐴 = +∞ ∧ 𝐵 = +∞) → 𝐴 = 𝐵)
3837orcd 407 . . . . . . 7 ((𝐴 = +∞ ∧ 𝐵 = +∞) → (𝐴 = 𝐵𝐵 < 𝐴))
3938a1d 25 . . . . . 6 ((𝐴 = +∞ ∧ 𝐵 = +∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
40 mnfltpnf 11904 . . . . . . . . . 10 -∞ < +∞
41 breq12 4618 . . . . . . . . . 10 ((𝐵 = -∞ ∧ 𝐴 = +∞) → (𝐵 < 𝐴 ↔ -∞ < +∞))
4240, 41mpbiri 248 . . . . . . . . 9 ((𝐵 = -∞ ∧ 𝐴 = +∞) → 𝐵 < 𝐴)
4342ancoms 469 . . . . . . . 8 ((𝐴 = +∞ ∧ 𝐵 = -∞) → 𝐵 < 𝐴)
4443olcd 408 . . . . . . 7 ((𝐴 = +∞ ∧ 𝐵 = -∞) → (𝐴 = 𝐵𝐵 < 𝐴))
4544a1d 25 . . . . . 6 ((𝐴 = +∞ ∧ 𝐵 = -∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
4636, 39, 453jaodan 1391 . . . . 5 ((𝐴 = +∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
47 mnflt 11901 . . . . . . . . 9 (𝐵 ∈ ℝ → -∞ < 𝐵)
4847adantl 482 . . . . . . . 8 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → -∞ < 𝐵)
49 breq1 4616 . . . . . . . . 9 (𝐴 = -∞ → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
5049adantr 481 . . . . . . . 8 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
5148, 50mpbird 247 . . . . . . 7 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → 𝐴 < 𝐵)
5251pm2.24d 147 . . . . . 6 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
53 breq12 4618 . . . . . . . 8 ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 ↔ -∞ < +∞))
5440, 53mpbiri 248 . . . . . . 7 ((𝐴 = -∞ ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
5554pm2.24d 147 . . . . . 6 ((𝐴 = -∞ ∧ 𝐵 = +∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
56 eqtr3 2642 . . . . . . . 8 ((𝐴 = -∞ ∧ 𝐵 = -∞) → 𝐴 = 𝐵)
5756orcd 407 . . . . . . 7 ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 = 𝐵𝐵 < 𝐴))
5857a1d 25 . . . . . 6 ((𝐴 = -∞ ∧ 𝐵 = -∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
5952, 55, 583jaodan 1391 . . . . 5 ((𝐴 = -∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
6029, 46, 593jaoian 1390 . . . 4 (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
6111, 12, 60syl2anb 496 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
6210, 61impbid 202 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴 = 𝐵𝐵 < 𝐴) ↔ ¬ 𝐴 < 𝐵))
6362con2bid 344 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3o 1035   = wceq 1480  wcel 1987   class class class wbr 4613  cr 9879  +∞cpnf 10015  -∞cmnf 10016  *cxr 10017   < clt 10018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-pre-lttri 9954  ax-pre-lttrn 9955
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-po 4995  df-so 4996  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023
This theorem is referenced by:  xrltso  11918  xrleloe  11921  xrltlen  11923
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