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Mirrors > Home > ILE Home > Th. List > renepnf | GIF version |
Description: No (finite) real equals plus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
Ref | Expression |
---|---|
renepnf | ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfnre 7432 | . . . 4 ⊢ +∞ ∉ ℝ | |
2 | 1 | neli 2346 | . . 3 ⊢ ¬ +∞ ∈ ℝ |
3 | eleq1 2145 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 ∈ ℝ ↔ +∞ ∈ ℝ)) | |
4 | 2, 3 | mtbiri 633 | . 2 ⊢ (𝐴 = +∞ → ¬ 𝐴 ∈ ℝ) |
5 | 4 | necon2ai 2303 | 1 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 ∈ wcel 1434 ≠ wne 2249 ℝcr 7252 +∞cpnf 7422 |
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-un 4224 ax-cnex 7339 ax-resscn 7340 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-rex 2359 df-rab 2362 df-v 2614 df-in 2990 df-ss 2997 df-pw 3408 df-uni 3628 df-pnf 7427 |
This theorem is referenced by: renepnfd 7441 renfdisj 7449 ltxrlt 7455 xrnepnf 9144 xrlttri3 9162 nltpnft 9174 xrrebnd 9176 rexneg 9187 |
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