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Theorem xrlttri 13059
Description: Ordering on the extended reals satisfies strict trichotomy. New proofs should generally use this instead of ax-pre-lttri 11126 or axlttri 11227. (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 13041 . . . . . . . 8 (𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴)
21adantr 482 . . . . . . 7 ((𝐴 ∈ ℝ*𝐴 = 𝐵) → ¬ 𝐴 < 𝐴)
3 breq2 5110 . . . . . . . 8 (𝐴 = 𝐵 → (𝐴 < 𝐴𝐴 < 𝐵))
43adantl 483 . . . . . . 7 ((𝐴 ∈ ℝ*𝐴 = 𝐵) → (𝐴 < 𝐴𝐴 < 𝐵))
52, 4mtbid 324 . . . . . 6 ((𝐴 ∈ ℝ*𝐴 = 𝐵) → ¬ 𝐴 < 𝐵)
65ex 414 . . . . 5 (𝐴 ∈ ℝ* → (𝐴 = 𝐵 → ¬ 𝐴 < 𝐵))
76adantr 482 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 = 𝐵 → ¬ 𝐴 < 𝐵))
8 xrltnsym 13057 . . . . 5 ((𝐵 ∈ ℝ*𝐴 ∈ ℝ*) → (𝐵 < 𝐴 → ¬ 𝐴 < 𝐵))
98ancoms 460 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐵 < 𝐴 → ¬ 𝐴 < 𝐵))
107, 9jaod 858 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴 = 𝐵𝐵 < 𝐴) → ¬ 𝐴 < 𝐵))
11 elxr 13038 . . . 4 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
12 elxr 13038 . . . 4 (𝐵 ∈ ℝ* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
13 axlttri 11227 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴)))
1413biimprd 248 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ (𝐴 = 𝐵𝐵 < 𝐴) → 𝐴 < 𝐵))
1514con1d 145 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
16 ltpnf 13042 . . . . . . . . 9 (𝐴 ∈ ℝ → 𝐴 < +∞)
1716adantr 482 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐴 < +∞)
18 breq2 5110 . . . . . . . . 9 (𝐵 = +∞ → (𝐴 < 𝐵𝐴 < +∞))
1918adantl 483 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 < 𝐵𝐴 < +∞))
2017, 19mpbird 257 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
2120pm2.24d 151 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
22 mnflt 13045 . . . . . . . . . 10 (𝐴 ∈ ℝ → -∞ < 𝐴)
2322adantr 482 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → -∞ < 𝐴)
24 breq1 5109 . . . . . . . . . 10 (𝐵 = -∞ → (𝐵 < 𝐴 ↔ -∞ < 𝐴))
2524adantl 483 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐵 < 𝐴 ↔ -∞ < 𝐴))
2623, 25mpbird 257 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → 𝐵 < 𝐴)
2726olcd 873 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 = 𝐵𝐵 < 𝐴))
2827a1d 25 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
2915, 21, 283jaodan 1431 . . . . 5 ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
30 ltpnf 13042 . . . . . . . . . 10 (𝐵 ∈ ℝ → 𝐵 < +∞)
3130adantl 483 . . . . . . . . 9 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → 𝐵 < +∞)
32 breq2 5110 . . . . . . . . . 10 (𝐴 = +∞ → (𝐵 < 𝐴𝐵 < +∞))
3332adantr 482 . . . . . . . . 9 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴𝐵 < +∞))
3431, 33mpbird 257 . . . . . . . 8 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → 𝐵 < 𝐴)
3534olcd 873 . . . . . . 7 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵𝐵 < 𝐴))
3635a1d 25 . . . . . 6 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
37 eqtr3 2763 . . . . . . . 8 ((𝐴 = +∞ ∧ 𝐵 = +∞) → 𝐴 = 𝐵)
3837orcd 872 . . . . . . 7 ((𝐴 = +∞ ∧ 𝐵 = +∞) → (𝐴 = 𝐵𝐵 < 𝐴))
3938a1d 25 . . . . . 6 ((𝐴 = +∞ ∧ 𝐵 = +∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
40 mnfltpnf 13048 . . . . . . . . . 10 -∞ < +∞
41 breq12 5111 . . . . . . . . . 10 ((𝐵 = -∞ ∧ 𝐴 = +∞) → (𝐵 < 𝐴 ↔ -∞ < +∞))
4240, 41mpbiri 258 . . . . . . . . 9 ((𝐵 = -∞ ∧ 𝐴 = +∞) → 𝐵 < 𝐴)
4342ancoms 460 . . . . . . . 8 ((𝐴 = +∞ ∧ 𝐵 = -∞) → 𝐵 < 𝐴)
4443olcd 873 . . . . . . 7 ((𝐴 = +∞ ∧ 𝐵 = -∞) → (𝐴 = 𝐵𝐵 < 𝐴))
4544a1d 25 . . . . . 6 ((𝐴 = +∞ ∧ 𝐵 = -∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
4636, 39, 453jaodan 1431 . . . . 5 ((𝐴 = +∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
47 mnflt 13045 . . . . . . . . 9 (𝐵 ∈ ℝ → -∞ < 𝐵)
4847adantl 483 . . . . . . . 8 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → -∞ < 𝐵)
49 breq1 5109 . . . . . . . . 9 (𝐴 = -∞ → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
5049adantr 482 . . . . . . . 8 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
5148, 50mpbird 257 . . . . . . 7 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → 𝐴 < 𝐵)
5251pm2.24d 151 . . . . . 6 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
53 breq12 5111 . . . . . . . 8 ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 ↔ -∞ < +∞))
5440, 53mpbiri 258 . . . . . . 7 ((𝐴 = -∞ ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
5554pm2.24d 151 . . . . . 6 ((𝐴 = -∞ ∧ 𝐵 = +∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
56 eqtr3 2763 . . . . . . . 8 ((𝐴 = -∞ ∧ 𝐵 = -∞) → 𝐴 = 𝐵)
5756orcd 872 . . . . . . 7 ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 = 𝐵𝐵 < 𝐴))
5857a1d 25 . . . . . 6 ((𝐴 = -∞ ∧ 𝐵 = -∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
5952, 55, 583jaodan 1431 . . . . 5 ((𝐴 = -∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
6029, 46, 593jaoian 1430 . . . 4 (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
6111, 12, 60syl2anb 599 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
6210, 61impbid 211 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴 = 𝐵𝐵 < 𝐴) ↔ ¬ 𝐴 < 𝐵))
6362con2bid 355 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846  w3o 1087   = wceq 1542  wcel 2107   class class class wbr 5106  cr 11051  +∞cpnf 11187  -∞cmnf 11188  *cxr 11189   < clt 11190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11108  ax-resscn 11109  ax-pre-lttri 11126  ax-pre-lttrn 11127
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-po 5546  df-so 5547  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-er 8649  df-en 8885  df-dom 8886  df-sdom 8887  df-pnf 11192  df-mnf 11193  df-xr 11194  df-ltxr 11195
This theorem is referenced by:  xrltso  13061  xrleloe  13064  xrltlen  13066
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