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Theorem xrltnr 8471
Description: The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.)
Assertion
Ref Expression
xrltnr (A * → ¬ A < A)

Proof of Theorem xrltnr
StepHypRef Expression
1 elxr 8466 . 2 (A * ↔ (A A = +∞ A = -∞))
2 ltnr 6892 . . 3 (A ℝ → ¬ A < A)
3 pnfnre 6864 . . . . . . . . . 10 +∞ ∉ ℝ
43neli 2293 . . . . . . . . 9 ¬ +∞
54intnan 837 . . . . . . . 8 ¬ (+∞ +∞ ℝ)
65intnanr 838 . . . . . . 7 ¬ ((+∞ +∞ ℝ) +∞ < +∞)
7 pnfnemnf 8467 . . . . . . . . 9 +∞ ≠ -∞
87neii 2205 . . . . . . . 8 ¬ +∞ = -∞
98intnanr 838 . . . . . . 7 ¬ (+∞ = -∞ +∞ = +∞)
106, 9pm3.2ni 725 . . . . . 6 ¬ (((+∞ +∞ ℝ) +∞ < +∞) (+∞ = -∞ +∞ = +∞))
114intnanr 838 . . . . . . 7 ¬ (+∞ +∞ = +∞)
124intnan 837 . . . . . . 7 ¬ (+∞ = -∞ +∞ ℝ)
1311, 12pm3.2ni 725 . . . . . 6 ¬ ((+∞ +∞ = +∞) (+∞ = -∞ +∞ ℝ))
1410, 13pm3.2ni 725 . . . . 5 ¬ ((((+∞ +∞ ℝ) +∞ < +∞) (+∞ = -∞ +∞ = +∞)) ((+∞ +∞ = +∞) (+∞ = -∞ +∞ ℝ)))
15 pnfxr 8462 . . . . . 6 +∞ *
16 ltxr 8465 . . . . . 6 ((+∞ * +∞ *) → (+∞ < +∞ ↔ ((((+∞ +∞ ℝ) +∞ < +∞) (+∞ = -∞ +∞ = +∞)) ((+∞ +∞ = +∞) (+∞ = -∞ +∞ ℝ)))))
1715, 15, 16mp2an 402 . . . . 5 (+∞ < +∞ ↔ ((((+∞ +∞ ℝ) +∞ < +∞) (+∞ = -∞ +∞ = +∞)) ((+∞ +∞ = +∞) (+∞ = -∞ +∞ ℝ))))
1814, 17mtbir 595 . . . 4 ¬ +∞ < +∞
19 breq12 3760 . . . . 5 ((A = +∞ A = +∞) → (A < A ↔ +∞ < +∞))
2019anidms 377 . . . 4 (A = +∞ → (A < A ↔ +∞ < +∞))
2118, 20mtbiri 599 . . 3 (A = +∞ → ¬ A < A)
22 mnfnre 6865 . . . . . . . . . 10 -∞ ∉ ℝ
2322neli 2293 . . . . . . . . 9 ¬ -∞
2423intnan 837 . . . . . . . 8 ¬ (-∞ -∞ ℝ)
2524intnanr 838 . . . . . . 7 ¬ ((-∞ -∞ ℝ) -∞ < -∞)
267nesymi 2245 . . . . . . . 8 ¬ -∞ = +∞
2726intnan 837 . . . . . . 7 ¬ (-∞ = -∞ -∞ = +∞)
2825, 27pm3.2ni 725 . . . . . 6 ¬ (((-∞ -∞ ℝ) -∞ < -∞) (-∞ = -∞ -∞ = +∞))
2923intnanr 838 . . . . . . 7 ¬ (-∞ -∞ = +∞)
3023intnan 837 . . . . . . 7 ¬ (-∞ = -∞ -∞ ℝ)
3129, 30pm3.2ni 725 . . . . . 6 ¬ ((-∞ -∞ = +∞) (-∞ = -∞ -∞ ℝ))
3228, 31pm3.2ni 725 . . . . 5 ¬ ((((-∞ -∞ ℝ) -∞ < -∞) (-∞ = -∞ -∞ = +∞)) ((-∞ -∞ = +∞) (-∞ = -∞ -∞ ℝ)))
33 mnfxr 8464 . . . . . 6 -∞ *
34 ltxr 8465 . . . . . 6 ((-∞ * -∞ *) → (-∞ < -∞ ↔ ((((-∞ -∞ ℝ) -∞ < -∞) (-∞ = -∞ -∞ = +∞)) ((-∞ -∞ = +∞) (-∞ = -∞ -∞ ℝ)))))
3533, 33, 34mp2an 402 . . . . 5 (-∞ < -∞ ↔ ((((-∞ -∞ ℝ) -∞ < -∞) (-∞ = -∞ -∞ = +∞)) ((-∞ -∞ = +∞) (-∞ = -∞ -∞ ℝ))))
3632, 35mtbir 595 . . . 4 ¬ -∞ < -∞
37 breq12 3760 . . . . 5 ((A = -∞ A = -∞) → (A < A ↔ -∞ < -∞))
3837anidms 377 . . . 4 (A = -∞ → (A < A ↔ -∞ < -∞))
3936, 38mtbiri 599 . . 3 (A = -∞ → ¬ A < A)
402, 21, 393jaoi 1197 . 2 ((A A = +∞ A = -∞) → ¬ A < A)
411, 40sylbi 114 1 (A * → ¬ A < A)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628   w3o 883   = wceq 1242   wcel 1390   class class class wbr 3755  cr 6710   < cltrr 6715  +∞cpnf 6854  -∞cmnf 6855  *cxr 6856   < clt 6857
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-cnex 6774  ax-resscn 6775  ax-pre-ltirr 6795
This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-pnf 6859  df-mnf 6860  df-xr 6861  df-ltxr 6862
This theorem is referenced by:  xrltnsym  8484  xrltso  8487  xrlttri3  8488  xrleid  8490  xrltne  8499  nltpnft  8500  ngtmnft  8501  xrrebnd  8502  lbioog  8552  ubioog  8553
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