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Theorem nltpnft 10049
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 10011 . 2 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2 renepnf 8227 . . . . 5 (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
32neneqd 2423 . . . 4 (𝐴 ∈ ℝ → ¬ 𝐴 = +∞)
4 ltpnf 10015 . . . . 5 (𝐴 ∈ ℝ → 𝐴 < +∞)
5 notnot 634 . . . . 5 (𝐴 < +∞ → ¬ ¬ 𝐴 < +∞)
64, 5syl 14 . . . 4 (𝐴 ∈ ℝ → ¬ ¬ 𝐴 < +∞)
73, 62falsed 709 . . 3 (𝐴 ∈ ℝ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
8 id 19 . . . 4 (𝐴 = +∞ → 𝐴 = +∞)
9 pnfxr 8232 . . . . . 6 +∞ ∈ ℝ*
10 xrltnr 10014 . . . . . 6 (+∞ ∈ ℝ* → ¬ +∞ < +∞)
119, 10ax-mp 5 . . . . 5 ¬ +∞ < +∞
12 breq1 4091 . . . . 5 (𝐴 = +∞ → (𝐴 < +∞ ↔ +∞ < +∞))
1311, 12mtbiri 681 . . . 4 (𝐴 = +∞ → ¬ 𝐴 < +∞)
148, 132thd 175 . . 3 (𝐴 = +∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
15 mnfnepnf 8235 . . . . . 6 -∞ ≠ +∞
1615neii 2404 . . . . 5 ¬ -∞ = +∞
17 eqeq1 2238 . . . . 5 (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞))
1816, 17mtbiri 681 . . . 4 (𝐴 = -∞ → ¬ 𝐴 = +∞)
19 mnfltpnf 10020 . . . . . . 7 -∞ < +∞
20 breq1 4091 . . . . . . 7 (𝐴 = -∞ → (𝐴 < +∞ ↔ -∞ < +∞))
2119, 20mpbiri 168 . . . . . 6 (𝐴 = -∞ → 𝐴 < +∞)
2221necon3bi 2452 . . . . 5 𝐴 < +∞ → 𝐴 ≠ -∞)
2322necon2bi 2457 . . . 4 (𝐴 = -∞ → ¬ ¬ 𝐴 < +∞)
2418, 232falsed 709 . . 3 (𝐴 = -∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
257, 14, 243jaoi 1339 . 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 1003   = wceq 1397  wcel 2202   class class class wbr 4088  cr 8031  +∞cpnf 8211  -∞cmnf 8212  *cxr 8213   < clt 8214
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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-pre-ltirr 8144
This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219
This theorem is referenced by:  npnflt  10050  xgepnf  10051  xrmaxiflemlub  11810
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