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Theorem xrlttri 11848
 Description: Ordering on the extended reals satisfies strict trichotomy. New proofs should generally use this instead of ax-pre-lttri 9889 or axlttri 9988. (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 11829 . . . . . . . 8 (𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴)
21adantr 480 . . . . . . 7 ((𝐴 ∈ ℝ*𝐴 = 𝐵) → ¬ 𝐴 < 𝐴)
3 breq2 4587 . . . . . . . 8 (𝐴 = 𝐵 → (𝐴 < 𝐴𝐴 < 𝐵))
43adantl 481 . . . . . . 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 480 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 = 𝐵 → ¬ 𝐴 < 𝐵))
8 xrltnsym 11846 . . . . 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 11826 . . . 4 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
12 elxr 11826 . . . 4 (𝐵 ∈ ℝ* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
13 axlttri 9988 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴)))
1413biimprd 237 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ (𝐴 = 𝐵𝐵 < 𝐴) → 𝐴 < 𝐵))
1514con1d 138 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
16 ltpnf 11830 . . . . . . . . 9 (𝐴 ∈ ℝ → 𝐴 < +∞)
1716adantr 480 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐴 < +∞)
18 breq2 4587 . . . . . . . . 9 (𝐵 = +∞ → (𝐴 < 𝐵𝐴 < +∞))
1918adantl 481 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 < 𝐵𝐴 < +∞))
2017, 19mpbird 246 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
2120pm2.24d 146 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
22 mnflt 11833 . . . . . . . . . 10 (𝐴 ∈ ℝ → -∞ < 𝐴)
2322adantr 480 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → -∞ < 𝐴)
24 breq1 4586 . . . . . . . . . 10 (𝐵 = -∞ → (𝐵 < 𝐴 ↔ -∞ < 𝐴))
2524adantl 481 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐵 < 𝐴 ↔ -∞ < 𝐴))
2623, 25mpbird 246 . . . . . . . 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 1386 . . . . 5 ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
30 ltpnf 11830 . . . . . . . . . 10 (𝐵 ∈ ℝ → 𝐵 < +∞)
3130adantl 481 . . . . . . . . 9 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → 𝐵 < +∞)
32 breq2 4587 . . . . . . . . . 10 (𝐴 = +∞ → (𝐵 < 𝐴𝐵 < +∞))
3332adantr 480 . . . . . . . . 9 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴𝐵 < +∞))
3431, 33mpbird 246 . . . . . . . 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 2631 . . . . . . . 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 11836 . . . . . . . . . 10 -∞ < +∞
41 breq12 4588 . . . . . . . . . 10 ((𝐵 = -∞ ∧ 𝐴 = +∞) → (𝐵 < 𝐴 ↔ -∞ < +∞))
4240, 41mpbiri 247 . . . . . . . . 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 1386 . . . . 5 ((𝐴 = +∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
47 mnflt 11833 . . . . . . . . 9 (𝐵 ∈ ℝ → -∞ < 𝐵)
4847adantl 481 . . . . . . . 8 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → -∞ < 𝐵)
49 breq1 4586 . . . . . . . . 9 (𝐴 = -∞ → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
5049adantr 480 . . . . . . . 8 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
5148, 50mpbird 246 . . . . . . 7 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → 𝐴 < 𝐵)
5251pm2.24d 146 . . . . . 6 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
53 breq12 4588 . . . . . . . 8 ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 ↔ -∞ < +∞))
5440, 53mpbiri 247 . . . . . . 7 ((𝐴 = -∞ ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
5554pm2.24d 146 . . . . . 6 ((𝐴 = -∞ ∧ 𝐵 = +∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
56 eqtr3 2631 . . . . . . . 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 1386 . . . . 5 ((𝐴 = -∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
6029, 46, 593jaoian 1385 . . . 4 (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
6111, 12, 60syl2anb 495 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵𝐵 < 𝐴)))
6210, 61impbid 201 . 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 195   ∨ wo 382   ∧ wa 383   ∨ w3o 1030   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ℝcr 9814  +∞cpnf 9950  -∞cmnf 9951  ℝ*cxr 9952   < clt 9953 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-pre-lttri 9889  ax-pre-lttrn 9890 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958 This theorem is referenced by:  xrltso  11850  xrleloe  11853  xrltlen  11855
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