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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 | pm5.61 795 |
. 2
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2 | elxr 9808 |
. . . 4
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3 | df-3or 981 |
. . . 4
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4 | or32 771 |
. . . 4
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5 | 2, 3, 4 | 3bitri 206 |
. . 3
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6 | df-ne 2361 |
. . 3
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7 | 5, 6 | anbi12i 460 |
. 2
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8 | renepnf 8036 |
. . . . 5
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9 | mnfnepnf 8044 |
. . . . . 6
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10 | neeq1 2373 |
. . . . . 6
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11 | 9, 10 | mpbiri 168 |
. . . . 5
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12 | 8, 11 | jaoi 717 |
. . . 4
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13 | 12 | neneqd 2381 |
. . 3
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14 | 13 | pm4.71i 391 |
. 2
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15 | 1, 7, 14 | 3bitr4i 212 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-un 4451 ax-cnex 7933 ax-resscn 7934 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-rex 2474 df-rab 2477 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-uni 3825 df-pnf 8025 df-mnf 8026 df-xr 8027 |
This theorem is referenced by: xaddnepnf 9890 |
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