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Mirrors > Home > ILE Home > Th. List > pnfnemnf | Unicode version |
Description: Plus and minus infinity are different elements of . (Contributed by NM, 14-Oct-2005.) |
Ref | Expression |
---|---|
pnfnemnf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfxr 7942 | . . . 4 | |
2 | pwne 4133 | . . . 4 | |
3 | 1, 2 | ax-mp 5 | . . 3 |
4 | 3 | necomi 2419 | . 2 |
5 | df-mnf 7927 | . 2 | |
6 | 4, 5 | neeqtrri 2363 | 1 |
Colors of variables: wff set class |
Syntax hints: wcel 2135 wne 2334 cpw 3553 cpnf 7921 cmnf 7922 cxr 7923 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-un 4405 ax-cnex 7835 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-rex 2448 df-rab 2451 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-uni 3784 df-pnf 7926 df-mnf 7927 df-xr 7928 |
This theorem is referenced by: mnfnepnf 7945 xnn0nemnf 9179 xrnemnf 9704 xrltnr 9706 pnfnlt 9714 nltmnf 9715 ngtmnft 9744 xrmnfdc 9770 xaddpnf1 9773 xaddnemnf 9784 xposdif 9809 xleaddadd 9814 |
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