![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > pnfge | GIF version |
Description: Plus infinity is an upper bound for extended reals. (Contributed by NM, 30-Jan-2006.) |
Ref | Expression |
---|---|
pnfge | ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfnlt 9460 | . 2 ⊢ (𝐴 ∈ ℝ* → ¬ +∞ < 𝐴) | |
2 | pnfxr 7736 | . . 3 ⊢ +∞ ∈ ℝ* | |
3 | xrlenlt 7747 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴 ≤ +∞ ↔ ¬ +∞ < 𝐴)) | |
4 | 2, 3 | mpan2 419 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 ≤ +∞ ↔ ¬ +∞ < 𝐴)) |
5 | 1, 4 | mpbird 166 | 1 ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ +∞) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 ∈ wcel 1461 class class class wbr 3893 +∞cpnf 7715 ℝ*cxr 7717 < clt 7718 ≤ cle 7719 |
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 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-cnex 7630 ax-resscn 7631 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-rab 2397 df-v 2657 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-br 3894 df-opab 3948 df-xp 4503 df-cnv 4505 df-pnf 7720 df-mnf 7721 df-xr 7722 df-ltxr 7723 df-le 7724 |
This theorem is referenced by: 0lepnf 9463 xrre2 9491 xleadd1a 9543 xltadd1 9546 xlt2add 9550 xsubge0 9551 xlesubadd 9553 xleaddadd 9557 elico2 9607 iccmax 9619 elxrge0 9648 xrmaxifle 10901 xrmaxadd 10916 xrbdtri 10931 isxmet2d 12331 blssec 12421 |
Copyright terms: Public domain | W3C validator |