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 9573 | . 2 ⊢ (𝐴 ∈ ℝ* → ¬ +∞ < 𝐴) | |
2 | pnfxr 7818 | . . 3 ⊢ +∞ ∈ ℝ* | |
3 | xrlenlt 7829 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴 ≤ +∞ ↔ ¬ +∞ < 𝐴)) | |
4 | 2, 3 | mpan2 421 | . 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 1480 class class class wbr 3929 +∞cpnf 7797 ℝ*cxr 7799 < clt 7800 ≤ cle 7801 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-cnex 7711 ax-resscn 7712 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-cnv 4547 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 |
This theorem is referenced by: 0lepnf 9576 xrre2 9604 xleadd1a 9656 xltadd1 9659 xlt2add 9663 xsubge0 9664 xlesubadd 9666 xleaddadd 9670 elico2 9720 iccmax 9732 elxrge0 9761 xrmaxifle 11015 xrmaxadd 11030 xrbdtri 11045 isxmet2d 12517 blssec 12607 |
Copyright terms: Public domain | W3C validator |