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Theorem xrltnsym 11955
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 11935 . 2 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2 elxr 11935 . 2 (𝐵 ∈ ℝ* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3 ltnsym 10120 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
4 rexr 10070 . . . . . . . 8 (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
5 pnfnlt 11947 . . . . . . . 8 (𝐴 ∈ ℝ* → ¬ +∞ < 𝐴)
64, 5syl 17 . . . . . . 7 (𝐴 ∈ ℝ → ¬ +∞ < 𝐴)
76adantr 481 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → ¬ +∞ < 𝐴)
8 breq1 4647 . . . . . . 7 (𝐵 = +∞ → (𝐵 < 𝐴 ↔ +∞ < 𝐴))
98adantl 482 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐵 < 𝐴 ↔ +∞ < 𝐴))
107, 9mtbird 315 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → ¬ 𝐵 < 𝐴)
1110a1d 25 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
12 nltmnf 11948 . . . . . . . 8 (𝐴 ∈ ℝ* → ¬ 𝐴 < -∞)
134, 12syl 17 . . . . . . 7 (𝐴 ∈ ℝ → ¬ 𝐴 < -∞)
1413adantr 481 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < -∞)
15 breq2 4648 . . . . . . 7 (𝐵 = -∞ → (𝐴 < 𝐵𝐴 < -∞))
1615adantl 482 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵𝐴 < -∞))
1714, 16mtbird 315 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < 𝐵)
1817pm2.21d 118 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
193, 11, 183jaodan 1392 . . 3 ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
20 pnfnlt 11947 . . . . . . 7 (𝐵 ∈ ℝ* → ¬ +∞ < 𝐵)
2120adantl 482 . . . . . 6 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ*) → ¬ +∞ < 𝐵)
22 breq1 4647 . . . . . . 7 (𝐴 = +∞ → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
2322adantr 481 . . . . . 6 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
2421, 23mtbird 315 . . . . 5 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ*) → ¬ 𝐴 < 𝐵)
2524pm2.21d 118 . . . 4 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
262, 25sylan2br 493 . . 3 ((𝐴 = +∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
27 rexr 10070 . . . . . . . 8 (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*)
28 nltmnf 11948 . . . . . . . 8 (𝐵 ∈ ℝ* → ¬ 𝐵 < -∞)
2927, 28syl 17 . . . . . . 7 (𝐵 ∈ ℝ → ¬ 𝐵 < -∞)
3029adantl 482 . . . . . 6 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → ¬ 𝐵 < -∞)
31 breq2 4648 . . . . . . 7 (𝐴 = -∞ → (𝐵 < 𝐴𝐵 < -∞))
3231adantr 481 . . . . . 6 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴𝐵 < -∞))
3330, 32mtbird 315 . . . . 5 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → ¬ 𝐵 < 𝐴)
3433a1d 25 . . . 4 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
35 mnfxr 10081 . . . . . . . 8 -∞ ∈ ℝ*
36 pnfnlt 11947 . . . . . . . 8 (-∞ ∈ ℝ* → ¬ +∞ < -∞)
3735, 36ax-mp 5 . . . . . . 7 ¬ +∞ < -∞
38 breq12 4649 . . . . . . 7 ((𝐵 = +∞ ∧ 𝐴 = -∞) → (𝐵 < 𝐴 ↔ +∞ < -∞))
3937, 38mtbiri 317 . . . . . 6 ((𝐵 = +∞ ∧ 𝐴 = -∞) → ¬ 𝐵 < 𝐴)
4039ancoms 469 . . . . 5 ((𝐴 = -∞ ∧ 𝐵 = +∞) → ¬ 𝐵 < 𝐴)
4140a1d 25 . . . 4 ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
42 xrltnr 11938 . . . . . . 7 (-∞ ∈ ℝ* → ¬ -∞ < -∞)
4335, 42ax-mp 5 . . . . . 6 ¬ -∞ < -∞
44 breq12 4649 . . . . . 6 ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ -∞ < -∞))
4543, 44mtbiri 317 . . . . 5 ((𝐴 = -∞ ∧ 𝐵 = -∞) → ¬ 𝐴 < 𝐵)
4645pm2.21d 118 . . . 4 ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
4734, 41, 463jaodan 1392 . . 3 ((𝐴 = -∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
4819, 26, 473jaoian 1391 . 2 (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
491, 2, 48syl2anb 496 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3o 1035   = wceq 1481  wcel 1988   class class class wbr 4644  cr 9920  +∞cpnf 10056  -∞cmnf 10057  *cxr 10058   < clt 10059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-pre-lttri 9995  ax-pre-lttrn 9996
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-po 5025  df-so 5026  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064
This theorem is referenced by:  xrltnsym2  11956  xrlttri  11957  xmullem2  12080  sgnp  13811  iccpartnel  41138
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