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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 9151 | . 2 ⊢ (𝐴 ∈ ℝ* → ¬ +∞ < 𝐴) | |
2 | pnfxr 7442 | . . 3 ⊢ +∞ ∈ ℝ* | |
3 | xrlenlt 7453 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴 ≤ +∞ ↔ ¬ +∞ < 𝐴)) | |
4 | 2, 3 | mpan2 416 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 ≤ +∞ ↔ ¬ +∞ < 𝐴)) |
5 | 1, 4 | mpbird 165 | 1 ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ +∞) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 103 ∈ wcel 1434 class class class wbr 3811 +∞cpnf 7421 ℝ*cxr 7423 < clt 7424 ≤ cle 7425 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-cnex 7338 ax-resscn 7339 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-rab 2362 df-v 2614 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-br 3812 df-opab 3866 df-xp 4406 df-cnv 4408 df-pnf 7426 df-mnf 7427 df-xr 7428 df-ltxr 7429 df-le 7430 |
This theorem is referenced by: 0lepnf 9154 xrre2 9177 elico2 9249 iccmax 9261 elxrge0 9290 |
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