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Theorem nltpnft 9490
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 9456 . 2 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2 renepnf 7737 . . . . 5 (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
32neneqd 2303 . . . 4 (𝐴 ∈ ℝ → ¬ 𝐴 = +∞)
4 ltpnf 9460 . . . . 5 (𝐴 ∈ ℝ → 𝐴 < +∞)
5 notnot 601 . . . . 5 (𝐴 < +∞ → ¬ ¬ 𝐴 < +∞)
64, 5syl 14 . . . 4 (𝐴 ∈ ℝ → ¬ ¬ 𝐴 < +∞)
73, 62falsed 674 . . 3 (𝐴 ∈ ℝ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
8 id 19 . . . 4 (𝐴 = +∞ → 𝐴 = +∞)
9 pnfxr 7742 . . . . . 6 +∞ ∈ ℝ*
10 xrltnr 9459 . . . . . 6 (+∞ ∈ ℝ* → ¬ +∞ < +∞)
119, 10ax-mp 7 . . . . 5 ¬ +∞ < +∞
12 breq1 3898 . . . . 5 (𝐴 = +∞ → (𝐴 < +∞ ↔ +∞ < +∞))
1311, 12mtbiri 647 . . . 4 (𝐴 = +∞ → ¬ 𝐴 < +∞)
148, 132thd 174 . . 3 (𝐴 = +∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
15 mnfnepnf 7745 . . . . . 6 -∞ ≠ +∞
1615neii 2284 . . . . 5 ¬ -∞ = +∞
17 eqeq1 2121 . . . . 5 (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞))
1816, 17mtbiri 647 . . . 4 (𝐴 = -∞ → ¬ 𝐴 = +∞)
19 mnfltpnf 9464 . . . . . . 7 -∞ < +∞
20 breq1 3898 . . . . . . 7 (𝐴 = -∞ → (𝐴 < +∞ ↔ -∞ < +∞))
2119, 20mpbiri 167 . . . . . 6 (𝐴 = -∞ → 𝐴 < +∞)
2221necon3bi 2332 . . . . 5 𝐴 < +∞ → 𝐴 ≠ -∞)
2322necon2bi 2337 . . . 4 (𝐴 = -∞ → ¬ ¬ 𝐴 < +∞)
2418, 232falsed 674 . . 3 (𝐴 = -∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
257, 14, 243jaoi 1264 . 2 ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
261, 25sylbi 120 1 (𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104  w3o 944   = wceq 1314  wcel 1463   class class class wbr 3895  cr 7546  +∞cpnf 7721  -∞cmnf 7722  *cxr 7723   < clt 7724
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-pre-ltirr 7657
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-xp 4505  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729
This theorem is referenced by:  npnflt  9491  xgepnf  9492  xrmaxiflemlub  10909
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