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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 9771 | . 2 ⊢ (𝐴 ∈ ℝ* → ¬ +∞ < 𝐴) | |
2 | pnfxr 7997 | . . 3 ⊢ +∞ ∈ ℝ* | |
3 | xrlenlt 8009 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴 ≤ +∞ ↔ ¬ +∞ < 𝐴)) | |
4 | 2, 3 | mpan2 425 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 ≤ +∞ ↔ ¬ +∞ < 𝐴)) |
5 | 1, 4 | mpbird 167 | 1 ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ +∞) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 105 ∈ wcel 2148 class class class wbr 4000 +∞cpnf 7976 ℝ*cxr 7978 < clt 7979 ≤ cle 7980 |
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 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-cnex 7890 ax-resscn 7891 |
This theorem depends on definitions: df-bi 117 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 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-xp 4629 df-cnv 4631 df-pnf 7981 df-mnf 7982 df-xr 7983 df-ltxr 7984 df-le 7985 |
This theorem is referenced by: 0lepnf 9774 xnn0dcle 9786 xnn0letri 9787 xrre2 9805 xleadd1a 9857 xltadd1 9860 xlt2add 9864 xsubge0 9865 xlesubadd 9867 xleaddadd 9871 elico2 9921 iccmax 9933 elxrge0 9962 elicore 10250 xrmaxifle 11235 xrmaxadd 11250 xrbdtri 11265 pcdvdsb 12299 pc2dvds 12309 pcaddlem 12318 isxmet2d 13508 blssec 13598 |
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