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Mirrors > Home > ILE Home > Th. List > pnfnlt | Unicode version |
Description: No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.) |
Ref | Expression |
---|---|
pnfnlt |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfnre 7997 |
. . . . . . 7
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2 | 1 | neli 2444 |
. . . . . 6
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3 | 2 | intnanr 930 |
. . . . 5
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4 | 3 | intnanr 930 |
. . . 4
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5 | pnfnemnf 8010 |
. . . . . 6
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6 | 5 | neii 2349 |
. . . . 5
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7 | 6 | intnanr 930 |
. . . 4
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8 | 4, 7 | pm3.2ni 813 |
. . 3
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9 | 2 | intnanr 930 |
. . . 4
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10 | 6 | intnanr 930 |
. . . 4
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11 | 9, 10 | pm3.2ni 813 |
. . 3
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12 | 8, 11 | pm3.2ni 813 |
. 2
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13 | pnfxr 8008 |
. . 3
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14 | ltxr 9773 |
. . 3
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15 | 13, 14 | mpan 424 |
. 2
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16 | 12, 15 | mtbiri 675 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-cnex 7901 ax-resscn 7902 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-xp 4632 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 |
This theorem is referenced by: pnfge 9787 xrltnsym 9791 xrlttr 9793 xrltso 9794 xltnegi 9833 xposdif 9880 qbtwnxr 10255 xrmaxiflemab 11250 xrmaxltsup 11261 |
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