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Mirrors > Home > ILE Home > Th. List > xrnepnf | Unicode version |
Description: An extended real other than plus infinity is real or negative infinite. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xrnepnf |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.61 766 |
. 2
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2 | elxr 9450 |
. . . 4
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3 | df-3or 944 |
. . . 4
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4 | or32 742 |
. . . 4
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5 | 2, 3, 4 | 3bitri 205 |
. . 3
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6 | df-ne 2281 |
. . 3
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7 | 5, 6 | anbi12i 453 |
. 2
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8 | renepnf 7731 |
. . . . 5
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9 | mnfnepnf 7739 |
. . . . . 6
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10 | neeq1 2293 |
. . . . . 6
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11 | 9, 10 | mpbiri 167 |
. . . . 5
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12 | 8, 11 | jaoi 688 |
. . . 4
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13 | 12 | neneqd 2301 |
. . 3
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14 | 13 | pm4.71i 386 |
. 2
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15 | 1, 7, 14 | 3bitr4i 211 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-un 4313 ax-cnex 7630 ax-resscn 7631 |
This theorem depends on definitions: df-bi 116 df-3or 944 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-rex 2394 df-rab 2397 df-v 2657 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-uni 3701 df-pnf 7720 df-mnf 7721 df-xr 7722 |
This theorem is referenced by: xaddnepnf 9528 |
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