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Theorem xrlttri 12573
 Description: Ordering on the extended reals satisfies strict trichotomy. New proofs should generally use this instead of ax-pre-lttri 10649 or axlttri 10750. (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 12555 . . . . . . . 8 (𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴)
21adantr 484 . . . . . . 7 ((𝐴 ∈ ℝ*𝐴 = 𝐵) → ¬ 𝐴 < 𝐴)
3 breq2 5036 . . . . . . . 8 (𝐴 = 𝐵 → (𝐴 < 𝐴𝐴 < 𝐵))
43adantl 485 . . . . . . 7 ((𝐴 ∈ ℝ*𝐴 = 𝐵) → (𝐴 < 𝐴𝐴 < 𝐵))
52, 4mtbid 327 . . . . . 6 ((𝐴 ∈ ℝ*𝐴 = 𝐵) → ¬ 𝐴 < 𝐵)
65ex 416 . . . . 5 (𝐴 ∈ ℝ* → (𝐴 = 𝐵 → ¬ 𝐴 < 𝐵))
76adantr 484 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 = 𝐵 → ¬ 𝐴 < 𝐵))
8 xrltnsym 12571 . . . . 5 ((𝐵 ∈ ℝ*𝐴 ∈ ℝ*) → (𝐵 < 𝐴 → ¬ 𝐴 < 𝐵))
98ancoms 462 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐵 < 𝐴 → ¬ 𝐴 < 𝐵))
107, 9jaod 856 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴 = 𝐵𝐵 < 𝐴) → ¬ 𝐴 < 𝐵))
11 elxr 12552 . . . 4 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
12 elxr 12552 . . . 4 (𝐵 ∈ ℝ* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
13 axlttri 10750 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴)))
1413biimprd 251 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ (𝐴 = 𝐵𝐵 < 𝐴) → 𝐴 < 𝐵))
1514con1d 147 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
16 ltpnf 12556 . . . . . . . . 9 (𝐴 ∈ ℝ → 𝐴 < +∞)
1716adantr 484 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐴 < +∞)
18 breq2 5036 . . . . . . . . 9 (𝐵 = +∞ → (𝐴 < 𝐵𝐴 < +∞))
1918adantl 485 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 < 𝐵𝐴 < +∞))
2017, 19mpbird 260 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
2120pm2.24d 154 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
22 mnflt 12559 . . . . . . . . . 10 (𝐴 ∈ ℝ → -∞ < 𝐴)
2322adantr 484 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → -∞ < 𝐴)
24 breq1 5035 . . . . . . . . . 10 (𝐵 = -∞ → (𝐵 < 𝐴 ↔ -∞ < 𝐴))
2524adantl 485 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐵 < 𝐴 ↔ -∞ < 𝐴))
2623, 25mpbird 260 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → 𝐵 < 𝐴)
2726olcd 871 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 = 𝐵𝐵 < 𝐴))
2827a1d 25 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
2915, 21, 283jaodan 1427 . . . . 5 ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
30 ltpnf 12556 . . . . . . . . . 10 (𝐵 ∈ ℝ → 𝐵 < +∞)
3130adantl 485 . . . . . . . . 9 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → 𝐵 < +∞)
32 breq2 5036 . . . . . . . . . 10 (𝐴 = +∞ → (𝐵 < 𝐴𝐵 < +∞))
3332adantr 484 . . . . . . . . 9 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴𝐵 < +∞))
3431, 33mpbird 260 . . . . . . . 8 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → 𝐵 < 𝐴)
3534olcd 871 . . . . . . 7 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵𝐵 < 𝐴))
3635a1d 25 . . . . . 6 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
37 eqtr3 2780 . . . . . . . 8 ((𝐴 = +∞ ∧ 𝐵 = +∞) → 𝐴 = 𝐵)
3837orcd 870 . . . . . . 7 ((𝐴 = +∞ ∧ 𝐵 = +∞) → (𝐴 = 𝐵𝐵 < 𝐴))
3938a1d 25 . . . . . 6 ((𝐴 = +∞ ∧ 𝐵 = +∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
40 mnfltpnf 12562 . . . . . . . . . 10 -∞ < +∞
41 breq12 5037 . . . . . . . . . 10 ((𝐵 = -∞ ∧ 𝐴 = +∞) → (𝐵 < 𝐴 ↔ -∞ < +∞))
4240, 41mpbiri 261 . . . . . . . . 9 ((𝐵 = -∞ ∧ 𝐴 = +∞) → 𝐵 < 𝐴)
4342ancoms 462 . . . . . . . 8 ((𝐴 = +∞ ∧ 𝐵 = -∞) → 𝐵 < 𝐴)
4443olcd 871 . . . . . . 7 ((𝐴 = +∞ ∧ 𝐵 = -∞) → (𝐴 = 𝐵𝐵 < 𝐴))
4544a1d 25 . . . . . 6 ((𝐴 = +∞ ∧ 𝐵 = -∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
4636, 39, 453jaodan 1427 . . . . 5 ((𝐴 = +∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
47 mnflt 12559 . . . . . . . . 9 (𝐵 ∈ ℝ → -∞ < 𝐵)
4847adantl 485 . . . . . . . 8 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → -∞ < 𝐵)
49 breq1 5035 . . . . . . . . 9 (𝐴 = -∞ → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
5049adantr 484 . . . . . . . 8 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
5148, 50mpbird 260 . . . . . . 7 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → 𝐴 < 𝐵)
5251pm2.24d 154 . . . . . 6 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
53 breq12 5037 . . . . . . . 8 ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 ↔ -∞ < +∞))
5440, 53mpbiri 261 . . . . . . 7 ((𝐴 = -∞ ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
5554pm2.24d 154 . . . . . 6 ((𝐴 = -∞ ∧ 𝐵 = +∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
56 eqtr3 2780 . . . . . . . 8 ((𝐴 = -∞ ∧ 𝐵 = -∞) → 𝐴 = 𝐵)
5756orcd 870 . . . . . . 7 ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 = 𝐵𝐵 < 𝐴))
5857a1d 25 . . . . . 6 ((𝐴 = -∞ ∧ 𝐵 = -∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
5952, 55, 583jaodan 1427 . . . . 5 ((𝐴 = -∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
6029, 46, 593jaoian 1426 . . . 4 (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
6111, 12, 60syl2anb 600 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
6210, 61impbid 215 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴 = 𝐵𝐵 < 𝐴) ↔ ¬ 𝐴 < 𝐵))
6362con2bid 358 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∨ w3o 1083   = wceq 1538   ∈ wcel 2111   class class class wbr 5032  ℝcr 10574  +∞cpnf 10710  -∞cmnf 10711  ℝ*cxr 10712   < clt 10713 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-pre-lttri 10649  ax-pre-lttrn 10650 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-po 5443  df-so 5444  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718 This theorem is referenced by:  xrltso  12575  xrleloe  12578  xrltlen  12580
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