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Theorem xrltnsym 8714
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 8696 . 2 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2 elxr 8696 . 2 (𝐵 ∈ ℝ* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3 ltnsym 7104 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
4 rexr 7071 . . . . . . . 8 (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
5 pnfnlt 8708 . . . . . . . 8 (𝐴 ∈ ℝ* → ¬ +∞ < 𝐴)
64, 5syl 14 . . . . . . 7 (𝐴 ∈ ℝ → ¬ +∞ < 𝐴)
76adantr 261 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → ¬ +∞ < 𝐴)
8 breq1 3767 . . . . . . 7 (𝐵 = +∞ → (𝐵 < 𝐴 ↔ +∞ < 𝐴))
98adantl 262 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐵 < 𝐴 ↔ +∞ < 𝐴))
107, 9mtbird 598 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → ¬ 𝐵 < 𝐴)
1110a1d 22 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
12 nltmnf 8709 . . . . . . . 8 (𝐴 ∈ ℝ* → ¬ 𝐴 < -∞)
134, 12syl 14 . . . . . . 7 (𝐴 ∈ ℝ → ¬ 𝐴 < -∞)
1413adantr 261 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < -∞)
15 breq2 3768 . . . . . . 7 (𝐵 = -∞ → (𝐴 < 𝐵𝐴 < -∞))
1615adantl 262 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵𝐴 < -∞))
1714, 16mtbird 598 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < 𝐵)
1817pm2.21d 549 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
193, 11, 183jaodan 1201 . . 3 ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
20 pnfnlt 8708 . . . . . . 7 (𝐵 ∈ ℝ* → ¬ +∞ < 𝐵)
2120adantl 262 . . . . . 6 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ*) → ¬ +∞ < 𝐵)
22 breq1 3767 . . . . . . 7 (𝐴 = +∞ → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
2322adantr 261 . . . . . 6 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
2421, 23mtbird 598 . . . . 5 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ*) → ¬ 𝐴 < 𝐵)
2524pm2.21d 549 . . . 4 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
262, 25sylan2br 272 . . 3 ((𝐴 = +∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
27 rexr 7071 . . . . . . . 8 (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*)
28 nltmnf 8709 . . . . . . . 8 (𝐵 ∈ ℝ* → ¬ 𝐵 < -∞)
2927, 28syl 14 . . . . . . 7 (𝐵 ∈ ℝ → ¬ 𝐵 < -∞)
3029adantl 262 . . . . . 6 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → ¬ 𝐵 < -∞)
31 breq2 3768 . . . . . . 7 (𝐴 = -∞ → (𝐵 < 𝐴𝐵 < -∞))
3231adantr 261 . . . . . 6 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴𝐵 < -∞))
3330, 32mtbird 598 . . . . 5 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → ¬ 𝐵 < 𝐴)
3433a1d 22 . . . 4 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
35 mnfxr 8694 . . . . . . . 8 -∞ ∈ ℝ*
36 pnfnlt 8708 . . . . . . . 8 (-∞ ∈ ℝ* → ¬ +∞ < -∞)
3735, 36ax-mp 7 . . . . . . 7 ¬ +∞ < -∞
38 breq12 3769 . . . . . . 7 ((𝐵 = +∞ ∧ 𝐴 = -∞) → (𝐵 < 𝐴 ↔ +∞ < -∞))
3937, 38mtbiri 600 . . . . . 6 ((𝐵 = +∞ ∧ 𝐴 = -∞) → ¬ 𝐵 < 𝐴)
4039ancoms 255 . . . . 5 ((𝐴 = -∞ ∧ 𝐵 = +∞) → ¬ 𝐵 < 𝐴)
4140a1d 22 . . . 4 ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
42 xrltnr 8701 . . . . . . 7 (-∞ ∈ ℝ* → ¬ -∞ < -∞)
4335, 42ax-mp 7 . . . . . 6 ¬ -∞ < -∞
44 breq12 3769 . . . . . 6 ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ -∞ < -∞))
4543, 44mtbiri 600 . . . . 5 ((𝐴 = -∞ ∧ 𝐵 = -∞) → ¬ 𝐴 < 𝐵)
4645pm2.21d 549 . . . 4 ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
4734, 41, 463jaodan 1201 . . 3 ((𝐴 = -∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
4819, 26, 473jaoian 1200 . 2 (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
491, 2, 48syl2anb 275 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97  wb 98  w3o 884   = wceq 1243  wcel 1393   class class class wbr 3764  cr 6888  +∞cpnf 7057  -∞cmnf 7058  *cxr 7059   < clt 7060
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-pre-ltirr 6996  ax-pre-lttrn 6998
This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065
This theorem is referenced by:  xrltnsym2  8715  xrltle  8719
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