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Theorem xrltnsym 12175
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 12155 . 2 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2 elxr 12155 . 2 (𝐵 ∈ ℝ* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3 ltnsym 10337 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
4 rexr 10287 . . . . . . . 8 (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
5 pnfnlt 12167 . . . . . . . 8 (𝐴 ∈ ℝ* → ¬ +∞ < 𝐴)
64, 5syl 17 . . . . . . 7 (𝐴 ∈ ℝ → ¬ +∞ < 𝐴)
76adantr 466 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → ¬ +∞ < 𝐴)
8 breq1 4789 . . . . . . 7 (𝐵 = +∞ → (𝐵 < 𝐴 ↔ +∞ < 𝐴))
98adantl 467 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐵 < 𝐴 ↔ +∞ < 𝐴))
107, 9mtbird 314 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → ¬ 𝐵 < 𝐴)
1110a1d 25 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
12 nltmnf 12168 . . . . . . . 8 (𝐴 ∈ ℝ* → ¬ 𝐴 < -∞)
134, 12syl 17 . . . . . . 7 (𝐴 ∈ ℝ → ¬ 𝐴 < -∞)
1413adantr 466 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < -∞)
15 breq2 4790 . . . . . . 7 (𝐵 = -∞ → (𝐴 < 𝐵𝐴 < -∞))
1615adantl 467 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵𝐴 < -∞))
1714, 16mtbird 314 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < 𝐵)
1817pm2.21d 119 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
193, 11, 183jaodan 1542 . . 3 ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
20 pnfnlt 12167 . . . . . . 7 (𝐵 ∈ ℝ* → ¬ +∞ < 𝐵)
2120adantl 467 . . . . . 6 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ*) → ¬ +∞ < 𝐵)
22 breq1 4789 . . . . . . 7 (𝐴 = +∞ → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
2322adantr 466 . . . . . 6 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
2421, 23mtbird 314 . . . . 5 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ*) → ¬ 𝐴 < 𝐵)
2524pm2.21d 119 . . . 4 ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
262, 25sylan2br 582 . . 3 ((𝐴 = +∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
27 rexr 10287 . . . . . . . 8 (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*)
28 nltmnf 12168 . . . . . . . 8 (𝐵 ∈ ℝ* → ¬ 𝐵 < -∞)
2927, 28syl 17 . . . . . . 7 (𝐵 ∈ ℝ → ¬ 𝐵 < -∞)
3029adantl 467 . . . . . 6 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → ¬ 𝐵 < -∞)
31 breq2 4790 . . . . . . 7 (𝐴 = -∞ → (𝐵 < 𝐴𝐵 < -∞))
3231adantr 466 . . . . . 6 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴𝐵 < -∞))
3330, 32mtbird 314 . . . . 5 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → ¬ 𝐵 < 𝐴)
3433a1d 25 . . . 4 ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
35 mnfxr 10298 . . . . . . . 8 -∞ ∈ ℝ*
36 pnfnlt 12167 . . . . . . . 8 (-∞ ∈ ℝ* → ¬ +∞ < -∞)
3735, 36ax-mp 5 . . . . . . 7 ¬ +∞ < -∞
38 breq12 4791 . . . . . . 7 ((𝐵 = +∞ ∧ 𝐴 = -∞) → (𝐵 < 𝐴 ↔ +∞ < -∞))
3937, 38mtbiri 316 . . . . . 6 ((𝐵 = +∞ ∧ 𝐴 = -∞) → ¬ 𝐵 < 𝐴)
4039ancoms 455 . . . . 5 ((𝐴 = -∞ ∧ 𝐵 = +∞) → ¬ 𝐵 < 𝐴)
4140a1d 25 . . . 4 ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
42 xrltnr 12158 . . . . . . 7 (-∞ ∈ ℝ* → ¬ -∞ < -∞)
4335, 42ax-mp 5 . . . . . 6 ¬ -∞ < -∞
44 breq12 4791 . . . . . 6 ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ -∞ < -∞))
4543, 44mtbiri 316 . . . . 5 ((𝐴 = -∞ ∧ 𝐵 = -∞) → ¬ 𝐴 < 𝐵)
4645pm2.21d 119 . . . 4 ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
4734, 41, 463jaodan 1542 . . 3 ((𝐴 = -∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
4819, 26, 473jaoian 1541 . 2 (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
491, 2, 48syl2anb 585 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3o 1070   = wceq 1631  wcel 2145   class class class wbr 4786  cr 10137  +∞cpnf 10273  -∞cmnf 10274  *cxr 10275   < clt 10276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-pre-lttri 10212  ax-pre-lttrn 10213
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-po 5170  df-so 5171  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281
This theorem is referenced by:  xrltnsym2  12176  xrlttri  12177  xmullem2  12300  sgnp  14038  iccpartnel  41902
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