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Theorem nltpnft 9012
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 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))

Proof of Theorem nltpnft
StepHypRef Expression
1 elxr 8980 . 2 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2 renepnf 7280 . . . . 5 (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
32neneqd 2270 . . . 4 (𝐴 ∈ ℝ → ¬ 𝐴 = +∞)
4 ltpnf 8984 . . . . 5 (𝐴 ∈ ℝ → 𝐴 < +∞)
5 notnot 592 . . . . 5 (𝐴 < +∞ → ¬ ¬ 𝐴 < +∞)
64, 5syl 14 . . . 4 (𝐴 ∈ ℝ → ¬ ¬ 𝐴 < +∞)
73, 62falsed 651 . . 3 (𝐴 ∈ ℝ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
8 id 19 . . . 4 (𝐴 = +∞ → 𝐴 = +∞)
9 pnfxr 7285 . . . . . 6 +∞ ∈ ℝ*
10 xrltnr 8983 . . . . . 6 (+∞ ∈ ℝ* → ¬ +∞ < +∞)
119, 10ax-mp 7 . . . . 5 ¬ +∞ < +∞
12 breq1 3808 . . . . 5 (𝐴 = +∞ → (𝐴 < +∞ ↔ +∞ < +∞))
1311, 12mtbiri 633 . . . 4 (𝐴 = +∞ → ¬ 𝐴 < +∞)
148, 132thd 173 . . 3 (𝐴 = +∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
15 mnfnepnf 7288 . . . . . 6 -∞ ≠ +∞
1615neii 2251 . . . . 5 ¬ -∞ = +∞
17 eqeq1 2089 . . . . 5 (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞))
1816, 17mtbiri 633 . . . 4 (𝐴 = -∞ → ¬ 𝐴 = +∞)
19 mnfltpnf 8988 . . . . . . 7 -∞ < +∞
20 breq1 3808 . . . . . . 7 (𝐴 = -∞ → (𝐴 < +∞ ↔ -∞ < +∞))
2119, 20mpbiri 166 . . . . . 6 (𝐴 = -∞ → 𝐴 < +∞)
2221necon3bi 2299 . . . . 5 𝐴 < +∞ → 𝐴 ≠ -∞)
2322necon2bi 2304 . . . 4 (𝐴 = -∞ → ¬ ¬ 𝐴 < +∞)
2418, 232falsed 651 . . 3 (𝐴 = -∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
257, 14, 243jaoi 1235 . 2 ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
261, 25sylbi 119 1 (𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103  w3o 919   = wceq 1285  wcel 1434   class class class wbr 3805  cr 7094  +∞cpnf 7264  -∞cmnf 7265  *cxr 7266   < clt 7267
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-pre-ltirr 7202
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-xp 4397  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272
This theorem is referenced by: (None)
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