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Mirrors > Home > ILE Home > Th. List > mnfxr | Unicode version |
Description: Minus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
Ref | Expression |
---|---|
mnfxr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mnf 8026 |
. . . . 5
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2 | pnfex 8042 |
. . . . . 6
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3 | 2 | pwex 4201 |
. . . . 5
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4 | 1, 3 | eqeltri 2262 |
. . . 4
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5 | 4 | prid2 3714 |
. . 3
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6 | elun2 3318 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 5, 6 | ax-mp 5 |
. 2
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8 | df-xr 8027 |
. 2
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9 | 7, 8 | eleqtrri 2265 |
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-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 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rex 2474 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: elxr 9808 xrltnr 9811 mnflt 9815 mnfltpnf 9817 nltmnf 9820 mnfle 9824 xrltnsym 9825 xrlttri3 9829 ngtmnft 9849 xrrebnd 9851 xrre2 9853 xrre3 9854 ge0gtmnf 9855 xnegcl 9864 xltnegi 9867 xaddf 9876 xaddval 9877 xaddmnf1 9880 xaddmnf2 9881 pnfaddmnf 9882 mnfaddpnf 9883 xrex 9888 xltadd1 9908 xlt2add 9912 xsubge0 9913 xposdif 9914 xleaddadd 9919 elioc2 9968 elico2 9969 elicc2 9970 ioomax 9980 iccmax 9981 elioomnf 10000 unirnioo 10005 xrmaxadd 11304 reopnap 14515 blssioo 14522 tgioo 14523 |
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