ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xrltnr GIF version

Theorem xrltnr 8801
Description: The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.)
Assertion
Ref Expression
xrltnr (𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴)

Proof of Theorem xrltnr
StepHypRef Expression
1 elxr 8796 . 2 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2 ltnr 7153 . . 3 (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
3 pnfnre 7125 . . . . . . . . . 10 +∞ ∉ ℝ
43neli 2316 . . . . . . . . 9 ¬ +∞ ∈ ℝ
54intnan 849 . . . . . . . 8 ¬ (+∞ ∈ ℝ ∧ +∞ ∈ ℝ)
65intnanr 850 . . . . . . 7 ¬ ((+∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ +∞ < +∞)
7 pnfnemnf 8797 . . . . . . . . 9 +∞ ≠ -∞
87neii 2222 . . . . . . . 8 ¬ +∞ = -∞
98intnanr 850 . . . . . . 7 ¬ (+∞ = -∞ ∧ +∞ = +∞)
106, 9pm3.2ni 737 . . . . . 6 ¬ (((+∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ +∞ < +∞) ∨ (+∞ = -∞ ∧ +∞ = +∞))
114intnanr 850 . . . . . . 7 ¬ (+∞ ∈ ℝ ∧ +∞ = +∞)
124intnan 849 . . . . . . 7 ¬ (+∞ = -∞ ∧ +∞ ∈ ℝ)
1311, 12pm3.2ni 737 . . . . . 6 ¬ ((+∞ ∈ ℝ ∧ +∞ = +∞) ∨ (+∞ = -∞ ∧ +∞ ∈ ℝ))
1410, 13pm3.2ni 737 . . . . 5 ¬ ((((+∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ +∞ < +∞) ∨ (+∞ = -∞ ∧ +∞ = +∞)) ∨ ((+∞ ∈ ℝ ∧ +∞ = +∞) ∨ (+∞ = -∞ ∧ +∞ ∈ ℝ)))
15 pnfxr 8792 . . . . . 6 +∞ ∈ ℝ*
16 ltxr 8795 . . . . . 6 ((+∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) → (+∞ < +∞ ↔ ((((+∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ +∞ < +∞) ∨ (+∞ = -∞ ∧ +∞ = +∞)) ∨ ((+∞ ∈ ℝ ∧ +∞ = +∞) ∨ (+∞ = -∞ ∧ +∞ ∈ ℝ)))))
1715, 15, 16mp2an 410 . . . . 5 (+∞ < +∞ ↔ ((((+∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ +∞ < +∞) ∨ (+∞ = -∞ ∧ +∞ = +∞)) ∨ ((+∞ ∈ ℝ ∧ +∞ = +∞) ∨ (+∞ = -∞ ∧ +∞ ∈ ℝ))))
1814, 17mtbir 606 . . . 4 ¬ +∞ < +∞
19 breq12 3796 . . . . 5 ((𝐴 = +∞ ∧ 𝐴 = +∞) → (𝐴 < 𝐴 ↔ +∞ < +∞))
2019anidms 383 . . . 4 (𝐴 = +∞ → (𝐴 < 𝐴 ↔ +∞ < +∞))
2118, 20mtbiri 610 . . 3 (𝐴 = +∞ → ¬ 𝐴 < 𝐴)
22 mnfnre 7126 . . . . . . . . . 10 -∞ ∉ ℝ
2322neli 2316 . . . . . . . . 9 ¬ -∞ ∈ ℝ
2423intnan 849 . . . . . . . 8 ¬ (-∞ ∈ ℝ ∧ -∞ ∈ ℝ)
2524intnanr 850 . . . . . . 7 ¬ ((-∞ ∈ ℝ ∧ -∞ ∈ ℝ) ∧ -∞ < -∞)
267nesymi 2266 . . . . . . . 8 ¬ -∞ = +∞
2726intnan 849 . . . . . . 7 ¬ (-∞ = -∞ ∧ -∞ = +∞)
2825, 27pm3.2ni 737 . . . . . 6 ¬ (((-∞ ∈ ℝ ∧ -∞ ∈ ℝ) ∧ -∞ < -∞) ∨ (-∞ = -∞ ∧ -∞ = +∞))
2923intnanr 850 . . . . . . 7 ¬ (-∞ ∈ ℝ ∧ -∞ = +∞)
3023intnan 849 . . . . . . 7 ¬ (-∞ = -∞ ∧ -∞ ∈ ℝ)
3129, 30pm3.2ni 737 . . . . . 6 ¬ ((-∞ ∈ ℝ ∧ -∞ = +∞) ∨ (-∞ = -∞ ∧ -∞ ∈ ℝ))
3228, 31pm3.2ni 737 . . . . 5 ¬ ((((-∞ ∈ ℝ ∧ -∞ ∈ ℝ) ∧ -∞ < -∞) ∨ (-∞ = -∞ ∧ -∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ -∞ = +∞) ∨ (-∞ = -∞ ∧ -∞ ∈ ℝ)))
33 mnfxr 8794 . . . . . 6 -∞ ∈ ℝ*
34 ltxr 8795 . . . . . 6 ((-∞ ∈ ℝ* ∧ -∞ ∈ ℝ*) → (-∞ < -∞ ↔ ((((-∞ ∈ ℝ ∧ -∞ ∈ ℝ) ∧ -∞ < -∞) ∨ (-∞ = -∞ ∧ -∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ -∞ = +∞) ∨ (-∞ = -∞ ∧ -∞ ∈ ℝ)))))
3533, 33, 34mp2an 410 . . . . 5 (-∞ < -∞ ↔ ((((-∞ ∈ ℝ ∧ -∞ ∈ ℝ) ∧ -∞ < -∞) ∨ (-∞ = -∞ ∧ -∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ -∞ = +∞) ∨ (-∞ = -∞ ∧ -∞ ∈ ℝ))))
3632, 35mtbir 606 . . . 4 ¬ -∞ < -∞
37 breq12 3796 . . . . 5 ((𝐴 = -∞ ∧ 𝐴 = -∞) → (𝐴 < 𝐴 ↔ -∞ < -∞))
3837anidms 383 . . . 4 (𝐴 = -∞ → (𝐴 < 𝐴 ↔ -∞ < -∞))
3936, 38mtbiri 610 . . 3 (𝐴 = -∞ → ¬ 𝐴 < 𝐴)
402, 21, 393jaoi 1209 . 2 ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → ¬ 𝐴 < 𝐴)
411, 40sylbi 118 1 (𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  wo 639  w3o 895   = wceq 1259  wcel 1409   class class class wbr 3791  cr 6945   < cltrr 6950  +∞cpnf 7115  -∞cmnf 7116  *cxr 7117   < clt 7118
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-cnex 7032  ax-resscn 7033  ax-pre-ltirr 7053
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-xp 4378  df-pnf 7120  df-mnf 7121  df-xr 7122  df-ltxr 7123
This theorem is referenced by:  xrltnsym  8814  xrltso  8817  xrlttri3  8818  xrleid  8820  xrltne  8829  nltpnft  8830  ngtmnft  8831  xrrebnd  8832  lbioog  8882  ubioog  8883
  Copyright terms: Public domain W3C validator