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Theorem nltpnft 9846
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 9808 . 2 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2 renepnf 8036 . . . . 5 (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
32neneqd 2381 . . . 4 (𝐴 ∈ ℝ → ¬ 𝐴 = +∞)
4 ltpnf 9812 . . . . 5 (𝐴 ∈ ℝ → 𝐴 < +∞)
5 notnot 630 . . . . 5 (𝐴 < +∞ → ¬ ¬ 𝐴 < +∞)
64, 5syl 14 . . . 4 (𝐴 ∈ ℝ → ¬ ¬ 𝐴 < +∞)
73, 62falsed 703 . . 3 (𝐴 ∈ ℝ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
8 id 19 . . . 4 (𝐴 = +∞ → 𝐴 = +∞)
9 pnfxr 8041 . . . . . 6 +∞ ∈ ℝ*
10 xrltnr 9811 . . . . . 6 (+∞ ∈ ℝ* → ¬ +∞ < +∞)
119, 10ax-mp 5 . . . . 5 ¬ +∞ < +∞
12 breq1 4021 . . . . 5 (𝐴 = +∞ → (𝐴 < +∞ ↔ +∞ < +∞))
1311, 12mtbiri 676 . . . 4 (𝐴 = +∞ → ¬ 𝐴 < +∞)
148, 132thd 175 . . 3 (𝐴 = +∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
15 mnfnepnf 8044 . . . . . 6 -∞ ≠ +∞
1615neii 2362 . . . . 5 ¬ -∞ = +∞
17 eqeq1 2196 . . . . 5 (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞))
1816, 17mtbiri 676 . . . 4 (𝐴 = -∞ → ¬ 𝐴 = +∞)
19 mnfltpnf 9817 . . . . . . 7 -∞ < +∞
20 breq1 4021 . . . . . . 7 (𝐴 = -∞ → (𝐴 < +∞ ↔ -∞ < +∞))
2119, 20mpbiri 168 . . . . . 6 (𝐴 = -∞ → 𝐴 < +∞)
2221necon3bi 2410 . . . . 5 𝐴 < +∞ → 𝐴 ≠ -∞)
2322necon2bi 2415 . . . 4 (𝐴 = -∞ → ¬ ¬ 𝐴 < +∞)
2418, 232falsed 703 . . 3 (𝐴 = -∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
257, 14, 243jaoi 1314 . 2 ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
261, 25sylbi 121 1 (𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 105  w3o 979   = wceq 1364  wcel 2160   class class class wbr 4018  cr 7841  +∞cpnf 8020  -∞cmnf 8021  *cxr 8022   < clt 8023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-pre-ltirr 7954
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4650  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028
This theorem is referenced by:  npnflt  9847  xgepnf  9848  xrmaxiflemlub  11291
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