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Mirrors > Home > ILE Home > Th. List > pnfxr | Unicode version |
Description: Plus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.) |
Ref | Expression |
---|---|
pnfxr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun2 3299 |
. . 3
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2 | df-pnf 7993 |
. . . . 5
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3 | cnex 7934 |
. . . . . . 7
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4 | 3 | uniex 4437 |
. . . . . 6
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5 | 4 | pwex 4183 |
. . . . 5
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6 | 2, 5 | eqeltri 2250 |
. . . 4
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7 | 6 | prid1 3698 |
. . 3
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8 | 1, 7 | sselii 3152 |
. 2
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9 | df-xr 7995 |
. 2
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10 | 8, 9 | eleqtrri 2253 |
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 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-cnex 7901 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-uni 3810 df-pnf 7993 df-xr 7995 |
This theorem is referenced by: pnfex 8010 pnfnemnf 8011 xnn0xr 9243 xrltnr 9778 ltpnf 9779 mnfltpnf 9784 pnfnlt 9786 pnfge 9788 xrlttri3 9796 xnn0dcle 9801 nltpnft 9813 xgepnf 9815 xrrebnd 9818 xrre 9819 xrre2 9820 xnegcl 9831 xaddf 9843 xaddval 9844 xaddpnf1 9845 xaddpnf2 9846 pnfaddmnf 9849 mnfaddpnf 9850 xrex 9855 xaddass2 9869 xltadd1 9875 xlt2add 9879 xsubge0 9880 xposdif 9881 xleaddadd 9886 elioc2 9935 elico2 9936 elicc2 9937 ioomax 9947 iccmax 9948 ioopos 9949 elioopnf 9966 elicopnf 9968 unirnioo 9972 elxrge0 9977 dfrp2 10263 elicore 10266 hashinfom 10757 rexico 11229 xrmaxiflemcl 11252 xrmaxadd 11268 fprodge0 11644 fprodge1 11646 pcxcl 12310 pc2dvds 12328 pcadd 12338 xblpnfps 13834 xblpnf 13835 xblss2ps 13840 blssec 13874 blpnfctr 13875 reopnap 13974 blssioo 13981 |
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