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Theorem xrltnsym 9572
Description: Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.)
Assertion
Ref Expression
xrltnsym ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))

Proof of Theorem xrltnsym
StepHypRef Expression
1 elxr 9556 . 2 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2 elxr 9556 . 2 (𝐵 ∈ ℝ* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3 ltnsym 7843 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
4 rexr 7804 . . . . . . . 8 (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
5 pnfnlt 9566 . . . . . . . 8 (𝐴 ∈ ℝ* → ¬ +∞ < 𝐴)
64, 5syl 14 . . . . . . 7 (𝐴 ∈ ℝ → ¬ +∞ < 𝐴)
76adantr 274 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → ¬ +∞ < 𝐴)
8 breq1 3927 . . . . . . 7 (𝐵 = +∞ → (𝐵 < 𝐴 ↔ +∞ < 𝐴))
98adantl 275 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐵 < 𝐴 ↔ +∞ < 𝐴))
107, 9mtbird 662 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → ¬ 𝐵 < 𝐴)
1110a1d 22 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
12 nltmnf 9567 . . . . . . . 8 (𝐴 ∈ ℝ* → ¬ 𝐴 < -∞)
134, 12syl 14 . . . . . . 7 (𝐴 ∈ ℝ → ¬ 𝐴 < -∞)
1413adantr 274 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < -∞)
15 breq2 3928 . . . . . . 7 (𝐵 = -∞ → (𝐴 < 𝐵𝐴 < -∞))
1615adantl 275 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵𝐴 < -∞))
1714, 16mtbird 662 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < 𝐵)
1817pm2.21d 608 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
193, 11, 183jaodan 1284 . . 3 ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
20 pnfnlt 9566 . . . . . . 7 (𝐵 ∈ ℝ* → ¬ +∞ < 𝐵)
2120adantl 275 . . . . . 6 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ*) → ¬ +∞ < 𝐵)
22 breq1 3927 . . . . . . 7 (𝐴 = +∞ → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
2322adantr 274 . . . . . 6 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
2421, 23mtbird 662 . . . . 5 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ*) → ¬ 𝐴 < 𝐵)
2524pm2.21d 608 . . . 4 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
262, 25sylan2br 286 . . 3 ((𝐴 = +∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
27 rexr 7804 . . . . . . . 8 (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*)
28 nltmnf 9567 . . . . . . . 8 (𝐵 ∈ ℝ* → ¬ 𝐵 < -∞)
2927, 28syl 14 . . . . . . 7 (𝐵 ∈ ℝ → ¬ 𝐵 < -∞)
3029adantl 275 . . . . . 6 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → ¬ 𝐵 < -∞)
31 breq2 3928 . . . . . . 7 (𝐴 = -∞ → (𝐵 < 𝐴𝐵 < -∞))
3231adantr 274 . . . . . 6 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴𝐵 < -∞))
3330, 32mtbird 662 . . . . 5 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → ¬ 𝐵 < 𝐴)
3433a1d 22 . . . 4 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
35 mnfxr 7815 . . . . . . . 8 -∞ ∈ ℝ*
36 pnfnlt 9566 . . . . . . . 8 (-∞ ∈ ℝ* → ¬ +∞ < -∞)
3735, 36ax-mp 5 . . . . . . 7 ¬ +∞ < -∞
38 breq12 3929 . . . . . . 7 ((𝐵 = +∞ ∧ 𝐴 = -∞) → (𝐵 < 𝐴 ↔ +∞ < -∞))
3937, 38mtbiri 664 . . . . . 6 ((𝐵 = +∞ ∧ 𝐴 = -∞) → ¬ 𝐵 < 𝐴)
4039ancoms 266 . . . . 5 ((𝐴 = -∞ ∧ 𝐵 = +∞) → ¬ 𝐵 < 𝐴)
4140a1d 22 . . . 4 ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
42 xrltnr 9559 . . . . . . 7 (-∞ ∈ ℝ* → ¬ -∞ < -∞)
4335, 42ax-mp 5 . . . . . 6 ¬ -∞ < -∞
44 breq12 3929 . . . . . 6 ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ -∞ < -∞))
4543, 44mtbiri 664 . . . . 5 ((𝐴 = -∞ ∧ 𝐵 = -∞) → ¬ 𝐴 < 𝐵)
4645pm2.21d 608 . . . 4 ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
4734, 41, 463jaodan 1284 . . 3 ((𝐴 = -∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
4819, 26, 473jaoian 1283 . 2 (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
491, 2, 48syl2anb 289 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  w3o 961   = wceq 1331  wcel 1480   class class class wbr 3924  cr 7612  +∞cpnf 7790  -∞cmnf 7791  *cxr 7792   < clt 7793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-pre-ltirr 7725  ax-pre-lttrn 7727
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-xp 4540  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798
This theorem is referenced by:  xrltnsym2  9573  xrltle  9577
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