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Mirrors > Home > ILE Home > Th. List > renepnf | Unicode 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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfnre 7067 | . . . 4 | |
2 | 1 | neli 2299 | . . 3 |
3 | eleq1 2100 | . . 3 | |
4 | 2, 3 | mtbiri 600 | . 2 |
5 | 4 | necon2ai 2259 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1243 wcel 1393 wne 2204 cr 6888 cpnf 7057 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-un 4170 ax-cnex 6975 ax-resscn 6976 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-rex 2312 df-rab 2315 df-v 2559 df-in 2924 df-ss 2931 df-pw 3361 df-uni 3581 df-pnf 7062 |
This theorem is referenced by: renepnfd 7076 renfdisj 7079 ltxrlt 7085 xrnepnf 8700 xrlttri3 8718 nltpnft 8730 xrrebnd 8732 rexneg 8743 |
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