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Mirrors > Home > ILE Home > Th. List > xrmnfdc | Unicode version |
Description: An extended real is or is not minus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.) |
Ref | Expression |
---|---|
xrmnfdc |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9774 |
. 2
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2 | renemnf 8004 |
. . . . . 6
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3 | 2 | neneqd 2368 |
. . . . 5
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4 | 3 | olcd 734 |
. . . 4
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5 | df-dc 835 |
. . . 4
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6 | 4, 5 | sylibr 134 |
. . 3
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7 | pnfnemnf 8010 |
. . . . . . 7
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8 | 7 | neii 2349 |
. . . . . 6
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9 | eqeq1 2184 |
. . . . . 6
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10 | 8, 9 | mtbiri 675 |
. . . . 5
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11 | 10 | olcd 734 |
. . . 4
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12 | 11, 5 | sylibr 134 |
. . 3
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13 | orc 712 |
. . . 4
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14 | 13, 5 | sylibr 134 |
. . 3
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15 | 6, 12, 14 | 3jaoi 1303 |
. 2
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16 | 1, 15 | sylbi 121 |
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-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 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-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-uni 3810 df-pnf 7992 df-mnf 7993 df-xr 7994 |
This theorem is referenced by: xaddf 9842 xaddval 9843 xaddmnf1 9846 xaddcom 9859 xnegdi 9866 xpncan 9869 xleadd1a 9871 xsubge0 9879 xrmaxiflemcl 11248 xrmaxifle 11249 xrmaxiflemab 11250 xrmaxiflemlub 11251 xrmaxiflemcom 11252 xrmaxadd 11264 |
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