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Theorem xrlttri 12165
Description: Ordering on the extended reals satisfies strict trichotomy. New proofs should generally use this instead of ax-pre-lttri 10202 or axlttri 10301. (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 12146 . . . . . . . 8 (𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴)
21adantr 472 . . . . . . 7 ((𝐴 ∈ ℝ*𝐴 = 𝐵) → ¬ 𝐴 < 𝐴)
3 breq2 4808 . . . . . . . 8 (𝐴 = 𝐵 → (𝐴 < 𝐴𝐴 < 𝐵))
43adantl 473 . . . . . . 7 ((𝐴 ∈ ℝ*𝐴 = 𝐵) → (𝐴 < 𝐴𝐴 < 𝐵))
52, 4mtbid 313 . . . . . 6 ((𝐴 ∈ ℝ*𝐴 = 𝐵) → ¬ 𝐴 < 𝐵)
65ex 449 . . . . 5 (𝐴 ∈ ℝ* → (𝐴 = 𝐵 → ¬ 𝐴 < 𝐵))
76adantr 472 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 = 𝐵 → ¬ 𝐴 < 𝐵))
8 xrltnsym 12163 . . . . 5 ((𝐵 ∈ ℝ*𝐴 ∈ ℝ*) → (𝐵 < 𝐴 → ¬ 𝐴 < 𝐵))
98ancoms 468 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐵 < 𝐴 → ¬ 𝐴 < 𝐵))
107, 9jaod 394 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴 = 𝐵𝐵 < 𝐴) → ¬ 𝐴 < 𝐵))
11 elxr 12143 . . . 4 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
12 elxr 12143 . . . 4 (𝐵 ∈ ℝ* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
13 axlttri 10301 . . . . . . . 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 12147 . . . . . . . . 9 (𝐴 ∈ ℝ → 𝐴 < +∞)
1716adantr 472 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐴 < +∞)
18 breq2 4808 . . . . . . . . 9 (𝐵 = +∞ → (𝐴 < 𝐵𝐴 < +∞))
1918adantl 473 . . . . . . . 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 12150 . . . . . . . . . 10 (𝐴 ∈ ℝ → -∞ < 𝐴)
2322adantr 472 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → -∞ < 𝐴)
24 breq1 4807 . . . . . . . . . 10 (𝐵 = -∞ → (𝐵 < 𝐴 ↔ -∞ < 𝐴))
2524adantl 473 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐵 < 𝐴 ↔ -∞ < 𝐴))
2623, 25mpbird 247 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → 𝐵 < 𝐴)
2726olcd 407 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 = 𝐵𝐵 < 𝐴))
2827a1d 25 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
2915, 21, 283jaodan 1543 . . . . 5 ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
30 ltpnf 12147 . . . . . . . . . 10 (𝐵 ∈ ℝ → 𝐵 < +∞)
3130adantl 473 . . . . . . . . 9 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → 𝐵 < +∞)
32 breq2 4808 . . . . . . . . . 10 (𝐴 = +∞ → (𝐵 < 𝐴𝐵 < +∞))
3332adantr 472 . . . . . . . . 9 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴𝐵 < +∞))
3431, 33mpbird 247 . . . . . . . 8 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → 𝐵 < 𝐴)
3534olcd 407 . . . . . . 7 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵𝐵 < 𝐴))
3635a1d 25 . . . . . 6 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
37 eqtr3 2781 . . . . . . . 8 ((𝐴 = +∞ ∧ 𝐵 = +∞) → 𝐴 = 𝐵)
3837orcd 406 . . . . . . 7 ((𝐴 = +∞ ∧ 𝐵 = +∞) → (𝐴 = 𝐵𝐵 < 𝐴))
3938a1d 25 . . . . . 6 ((𝐴 = +∞ ∧ 𝐵 = +∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
40 mnfltpnf 12153 . . . . . . . . . 10 -∞ < +∞
41 breq12 4809 . . . . . . . . . 10 ((𝐵 = -∞ ∧ 𝐴 = +∞) → (𝐵 < 𝐴 ↔ -∞ < +∞))
4240, 41mpbiri 248 . . . . . . . . 9 ((𝐵 = -∞ ∧ 𝐴 = +∞) → 𝐵 < 𝐴)
4342ancoms 468 . . . . . . . 8 ((𝐴 = +∞ ∧ 𝐵 = -∞) → 𝐵 < 𝐴)
4443olcd 407 . . . . . . 7 ((𝐴 = +∞ ∧ 𝐵 = -∞) → (𝐴 = 𝐵𝐵 < 𝐴))
4544a1d 25 . . . . . 6 ((𝐴 = +∞ ∧ 𝐵 = -∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
4636, 39, 453jaodan 1543 . . . . 5 ((𝐴 = +∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
47 mnflt 12150 . . . . . . . . 9 (𝐵 ∈ ℝ → -∞ < 𝐵)
4847adantl 473 . . . . . . . 8 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → -∞ < 𝐵)
49 breq1 4807 . . . . . . . . 9 (𝐴 = -∞ → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
5049adantr 472 . . . . . . . 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 4809 . . . . . . . 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 2781 . . . . . . . 8 ((𝐴 = -∞ ∧ 𝐵 = -∞) → 𝐴 = 𝐵)
5756orcd 406 . . . . . . 7 ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 = 𝐵𝐵 < 𝐴))
5857a1d 25 . . . . . 6 ((𝐴 = -∞ ∧ 𝐵 = -∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
5952, 55, 583jaodan 1543 . . . . 5 ((𝐴 = -∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
6029, 46, 593jaoian 1542 . . . 4 (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
6111, 12, 60syl2anb 497 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
6210, 61impbid 202 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴 = 𝐵𝐵 < 𝐴) ↔ ¬ 𝐴 < 𝐵))
6362con2bid 343 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 382  wa 383  w3o 1071   = wceq 1632  wcel 2139   class class class wbr 4804  cr 10127  +∞cpnf 10263  -∞cmnf 10264  *cxr 10265   < clt 10266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-pre-lttri 10202  ax-pre-lttrn 10203
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271
This theorem is referenced by:  xrltso  12167  xrleloe  12170  xrltlen  12172
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