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Mirrors > Home > ILE Home > Th. List > xgepnf | GIF version |
Description: An extended real which is greater than plus infinity is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.) |
Ref | Expression |
---|---|
xgepnf | ⊢ (𝐴 ∈ ℝ* → (+∞ ≤ 𝐴 ↔ 𝐴 = +∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfxr 7925 | . . 3 ⊢ +∞ ∈ ℝ* | |
2 | xrlenlt 7937 | . . 3 ⊢ ((+∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (+∞ ≤ 𝐴 ↔ ¬ 𝐴 < +∞)) | |
3 | 1, 2 | mpan 421 | . 2 ⊢ (𝐴 ∈ ℝ* → (+∞ ≤ 𝐴 ↔ ¬ 𝐴 < +∞)) |
4 | nltpnft 9713 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) | |
5 | 3, 4 | bitr4d 190 | 1 ⊢ (𝐴 ∈ ℝ* → (+∞ ≤ 𝐴 ↔ 𝐴 = +∞)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 = wceq 1335 ∈ wcel 2128 class class class wbr 3965 +∞cpnf 7904 ℝ*cxr 7906 < clt 7907 ≤ cle 7908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-cnex 7818 ax-resscn 7819 ax-pre-ltirr 7839 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-xp 4591 df-cnv 4593 df-pnf 7909 df-mnf 7910 df-xr 7911 df-ltxr 7912 df-le 7913 |
This theorem is referenced by: xnn0lenn0nn0 9764 xleaddadd 9786 |
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