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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 | pm5.61 707 |
. 2
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2 | elxr 8466 |
. . . 4
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3 | df-3or 885 |
. . . 4
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4 | or32 686 |
. . . 4
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5 | 2, 3, 4 | 3bitri 195 |
. . 3
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6 | df-ne 2203 |
. . 3
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7 | 5, 6 | anbi12i 433 |
. 2
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8 | renepnf 6870 |
. . . . 5
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9 | mnfnepnf 8468 |
. . . . . 6
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10 | neeq1 2213 |
. . . . . 6
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11 | 9, 10 | mpbiri 157 |
. . . . 5
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12 | 8, 11 | jaoi 635 |
. . . 4
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13 | 12 | neneqd 2221 |
. . 3
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14 | 13 | pm4.71i 371 |
. 2
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15 | 1, 7, 14 | 3bitr4i 201 |
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 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-un 4136 ax-cnex 6774 ax-resscn 6775 |
This theorem depends on definitions: df-bi 110 df-3or 885 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-nel 2204 df-rex 2306 df-rab 2309 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-uni 3572 df-pnf 6859 df-mnf 6860 df-xr 6861 |
This theorem is referenced by: (None) |
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