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Theorem nltpnft 8480
Description: An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.)
Assertion
Ref Expression
nltpnft (A * → (A = +∞ ↔ ¬ A < +∞))

Proof of Theorem nltpnft
StepHypRef Expression
1 elxr 8446 . 2 (A * ↔ (A A = +∞ A = -∞))
2 renepnf 6850 . . . . 5 (A ℝ → A ≠ +∞)
32neneqd 2221 . . . 4 (A ℝ → ¬ A = +∞)
4 ltpnf 8452 . . . . 5 (A ℝ → A < +∞)
5 notnot1 559 . . . . 5 (A < +∞ → ¬ ¬ A < +∞)
64, 5syl 14 . . . 4 (A ℝ → ¬ ¬ A < +∞)
73, 62falsed 617 . . 3 (A ℝ → (A = +∞ ↔ ¬ A < +∞))
8 id 19 . . . 4 (A = +∞ → A = +∞)
9 pnfxr 8442 . . . . . 6 +∞ *
10 xrltnr 8451 . . . . . 6 (+∞ * → ¬ +∞ < +∞)
119, 10ax-mp 7 . . . . 5 ¬ +∞ < +∞
12 breq1 3758 . . . . 5 (A = +∞ → (A < +∞ ↔ +∞ < +∞))
1311, 12mtbiri 599 . . . 4 (A = +∞ → ¬ A < +∞)
148, 132thd 164 . . 3 (A = +∞ → (A = +∞ ↔ ¬ A < +∞))
15 mnfnepnf 8448 . . . . . 6 -∞ ≠ +∞
1615neii 2205 . . . . 5 ¬ -∞ = +∞
17 eqeq1 2043 . . . . 5 (A = -∞ → (A = +∞ ↔ -∞ = +∞))
1816, 17mtbiri 599 . . . 4 (A = -∞ → ¬ A = +∞)
19 mnfltpnf 8456 . . . . . . 7 -∞ < +∞
20 breq1 3758 . . . . . . 7 (A = -∞ → (A < +∞ ↔ -∞ < +∞))
2119, 20mpbiri 157 . . . . . 6 (A = -∞ → A < +∞)
2221necon3bi 2249 . . . . 5 A < +∞ → A ≠ -∞)
2322necon2bi 2254 . . . 4 (A = -∞ → ¬ ¬ A < +∞)
2418, 232falsed 617 . . 3 (A = -∞ → (A = +∞ ↔ ¬ A < +∞))
257, 14, 243jaoi 1197 . 2 ((A A = +∞ A = -∞) → (A = +∞ ↔ ¬ A < +∞))
261, 25sylbi 114 1 (A * → (A = +∞ ↔ ¬ A < +∞))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98   w3o 883   = wceq 1242   wcel 1390   class class class wbr 3755  cr 6690  +∞cpnf 6834  -∞cmnf 6835  *cxr 6836   < clt 6837
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-bnd 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 6754  ax-resscn 6755  ax-pre-ltirr 6775
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 6839  df-mnf 6840  df-xr 6841  df-ltxr 6842
This theorem is referenced by: (None)
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