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Theorem xrltnsym 8444
Description: Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.)
Assertion
Ref Expression
xrltnsym ((A * B *) → (A < B → ¬ B < A))

Proof of Theorem xrltnsym
StepHypRef Expression
1 elxr 8426 . 2 (A * ↔ (A A = +∞ A = -∞))
2 elxr 8426 . 2 (B * ↔ (B B = +∞ B = -∞))
3 ltnsym 6861 . . . 4 ((A B ℝ) → (A < B → ¬ B < A))
4 rexr 6828 . . . . . . . 8 (A ℝ → A *)
5 pnfnlt 8438 . . . . . . . 8 (A * → ¬ +∞ < A)
64, 5syl 14 . . . . . . 7 (A ℝ → ¬ +∞ < A)
76adantr 261 . . . . . 6 ((A B = +∞) → ¬ +∞ < A)
8 breq1 3758 . . . . . . 7 (B = +∞ → (B < A ↔ +∞ < A))
98adantl 262 . . . . . 6 ((A B = +∞) → (B < A ↔ +∞ < A))
107, 9mtbird 597 . . . . 5 ((A B = +∞) → ¬ B < A)
1110a1d 22 . . . 4 ((A B = +∞) → (A < B → ¬ B < A))
12 nltmnf 8439 . . . . . . . 8 (A * → ¬ A < -∞)
134, 12syl 14 . . . . . . 7 (A ℝ → ¬ A < -∞)
1413adantr 261 . . . . . 6 ((A B = -∞) → ¬ A < -∞)
15 breq2 3759 . . . . . . 7 (B = -∞ → (A < BA < -∞))
1615adantl 262 . . . . . 6 ((A B = -∞) → (A < BA < -∞))
1714, 16mtbird 597 . . . . 5 ((A B = -∞) → ¬ A < B)
1817pm2.21d 549 . . . 4 ((A B = -∞) → (A < B → ¬ B < A))
193, 11, 183jaodan 1200 . . 3 ((A (B B = +∞ B = -∞)) → (A < B → ¬ B < A))
20 pnfnlt 8438 . . . . . . 7 (B * → ¬ +∞ < B)
2120adantl 262 . . . . . 6 ((A = +∞ B *) → ¬ +∞ < B)
22 breq1 3758 . . . . . . 7 (A = +∞ → (A < B ↔ +∞ < B))
2322adantr 261 . . . . . 6 ((A = +∞ B *) → (A < B ↔ +∞ < B))
2421, 23mtbird 597 . . . . 5 ((A = +∞ B *) → ¬ A < B)
2524pm2.21d 549 . . . 4 ((A = +∞ B *) → (A < B → ¬ B < A))
262, 25sylan2br 272 . . 3 ((A = +∞ (B B = +∞ B = -∞)) → (A < B → ¬ B < A))
27 rexr 6828 . . . . . . . 8 (B ℝ → B *)
28 nltmnf 8439 . . . . . . . 8 (B * → ¬ B < -∞)
2927, 28syl 14 . . . . . . 7 (B ℝ → ¬ B < -∞)
3029adantl 262 . . . . . 6 ((A = -∞ B ℝ) → ¬ B < -∞)
31 breq2 3759 . . . . . . 7 (A = -∞ → (B < AB < -∞))
3231adantr 261 . . . . . 6 ((A = -∞ B ℝ) → (B < AB < -∞))
3330, 32mtbird 597 . . . . 5 ((A = -∞ B ℝ) → ¬ B < A)
3433a1d 22 . . . 4 ((A = -∞ B ℝ) → (A < B → ¬ B < A))
35 mnfxr 8424 . . . . . . . 8 -∞ *
36 pnfnlt 8438 . . . . . . . 8 (-∞ * → ¬ +∞ < -∞)
3735, 36ax-mp 7 . . . . . . 7 ¬ +∞ < -∞
38 breq12 3760 . . . . . . 7 ((B = +∞ A = -∞) → (B < A ↔ +∞ < -∞))
3937, 38mtbiri 599 . . . . . 6 ((B = +∞ A = -∞) → ¬ B < A)
4039ancoms 255 . . . . 5 ((A = -∞ B = +∞) → ¬ B < A)
4140a1d 22 . . . 4 ((A = -∞ B = +∞) → (A < B → ¬ B < A))
42 xrltnr 8431 . . . . . . 7 (-∞ * → ¬ -∞ < -∞)
4335, 42ax-mp 7 . . . . . 6 ¬ -∞ < -∞
44 breq12 3760 . . . . . 6 ((A = -∞ B = -∞) → (A < B ↔ -∞ < -∞))
4543, 44mtbiri 599 . . . . 5 ((A = -∞ B = -∞) → ¬ A < B)
4645pm2.21d 549 . . . 4 ((A = -∞ B = -∞) → (A < B → ¬ B < A))
4734, 41, 463jaodan 1200 . . 3 ((A = -∞ (B B = +∞ B = -∞)) → (A < B → ¬ B < A))
4819, 26, 473jaoian 1199 . 2 (((A A = +∞ A = -∞) (B B = +∞ B = -∞)) → (A < B → ¬ B < A))
491, 2, 48syl2anb 275 1 ((A * B *) → (A < B → ¬ B < A))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   w3o 883   = wceq 1242   wcel 1390   class class class wbr 3755  cr 6670  +∞cpnf 6814  -∞cmnf 6815  *cxr 6816   < clt 6817
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-cnex 6734  ax-resscn 6735  ax-pre-ltirr 6755  ax-pre-lttrn 6757
This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-pnf 6819  df-mnf 6820  df-xr 6821  df-ltxr 6822
This theorem is referenced by:  xrltnsym2  8445  xrltle  8449
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