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Theorem nltpnft 9814
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 9776 . 2 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2 renepnf 8005 . . . . 5 (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
32neneqd 2368 . . . 4 (𝐴 ∈ ℝ → ¬ 𝐴 = +∞)
4 ltpnf 9780 . . . . 5 (𝐴 ∈ ℝ → 𝐴 < +∞)
5 notnot 629 . . . . 5 (𝐴 < +∞ → ¬ ¬ 𝐴 < +∞)
64, 5syl 14 . . . 4 (𝐴 ∈ ℝ → ¬ ¬ 𝐴 < +∞)
73, 62falsed 702 . . 3 (𝐴 ∈ ℝ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
8 id 19 . . . 4 (𝐴 = +∞ → 𝐴 = +∞)
9 pnfxr 8010 . . . . . 6 +∞ ∈ ℝ*
10 xrltnr 9779 . . . . . 6 (+∞ ∈ ℝ* → ¬ +∞ < +∞)
119, 10ax-mp 5 . . . . 5 ¬ +∞ < +∞
12 breq1 4007 . . . . 5 (𝐴 = +∞ → (𝐴 < +∞ ↔ +∞ < +∞))
1311, 12mtbiri 675 . . . 4 (𝐴 = +∞ → ¬ 𝐴 < +∞)
148, 132thd 175 . . 3 (𝐴 = +∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
15 mnfnepnf 8013 . . . . . 6 -∞ ≠ +∞
1615neii 2349 . . . . 5 ¬ -∞ = +∞
17 eqeq1 2184 . . . . 5 (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞))
1816, 17mtbiri 675 . . . 4 (𝐴 = -∞ → ¬ 𝐴 = +∞)
19 mnfltpnf 9785 . . . . . . 7 -∞ < +∞
20 breq1 4007 . . . . . . 7 (𝐴 = -∞ → (𝐴 < +∞ ↔ -∞ < +∞))
2119, 20mpbiri 168 . . . . . 6 (𝐴 = -∞ → 𝐴 < +∞)
2221necon3bi 2397 . . . . 5 𝐴 < +∞ → 𝐴 ≠ -∞)
2322necon2bi 2402 . . . 4 (𝐴 = -∞ → ¬ ¬ 𝐴 < +∞)
2418, 232falsed 702 . . 3 (𝐴 = -∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
257, 14, 243jaoi 1303 . 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 977   = wceq 1353  wcel 2148   class class class wbr 4004  cr 7810  +∞cpnf 7989  -∞cmnf 7990  *cxr 7991   < clt 7992
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-pre-ltirr 7923
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-xp 4633  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997
This theorem is referenced by:  npnflt  9815  xgepnf  9816  xrmaxiflemlub  11256
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