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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 7527 | . . . 4 ⊢ +∞ ∉ ℝ | |
2 | 1 | neli 2352 | . . 3 ⊢ ¬ +∞ ∈ ℝ |
3 | eleq1 2150 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 ∈ ℝ ↔ +∞ ∈ ℝ)) | |
4 | 2, 3 | mtbiri 635 | . 2 ⊢ (𝐴 = +∞ → ¬ 𝐴 ∈ ℝ) |
5 | 4 | necon2ai 2309 | 1 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1289 ∈ wcel 1438 ≠ wne 2255 ℝcr 7347 +∞cpnf 7517 |
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 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-un 4260 ax-cnex 7434 ax-resscn 7435 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-rex 2365 df-rab 2368 df-v 2621 df-in 3005 df-ss 3012 df-pw 3431 df-uni 3654 df-pnf 7522 |
This theorem is referenced by: renepnfd 7536 renfdisj 7544 ltxrlt 7550 xrnepnf 9247 xrlttri3 9265 nltpnft 9277 xrrebnd 9279 rexneg 9290 |
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