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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 7831 | . . . 4 ⊢ +∞ ∉ ℝ | |
2 | 1 | neli 2406 | . . 3 ⊢ ¬ +∞ ∈ ℝ |
3 | eleq1 2203 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 ∈ ℝ ↔ +∞ ∈ ℝ)) | |
4 | 2, 3 | mtbiri 665 | . 2 ⊢ (𝐴 = +∞ → ¬ 𝐴 ∈ ℝ) |
5 | 4 | necon2ai 2363 | 1 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 ≠ wne 2309 ℝcr 7643 +∞cpnf 7821 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-un 4363 ax-cnex 7735 ax-resscn 7736 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-rex 2423 df-rab 2426 df-v 2691 df-in 3082 df-ss 3089 df-pw 3517 df-uni 3745 df-pnf 7826 |
This theorem is referenced by: renepnfd 7840 renfdisj 7848 ltxrlt 7854 xrnepnf 9595 xrlttri3 9613 nltpnft 9627 xrrebnd 9632 rexneg 9643 xrpnfdc 9655 rexadd 9665 xaddnepnf 9671 xaddcom 9674 xaddid1 9675 xnn0xadd0 9680 xnegdi 9681 xpncan 9684 xleadd1a 9686 xltadd1 9689 xsubge0 9694 xposdif 9695 xleaddadd 9700 xrmaxrecl 11056 |
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