Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mnf 7063 | . . . . 5 | |
2 | pnfex 8693 | . . . . . 6 | |
3 | 2 | pwex 3932 | . . . . 5 |
4 | 1, 3 | eqeltri 2110 | . . . 4 |
5 | 4 | prid2 3477 | . . 3 |
6 | elun2 3111 | . . 3 | |
7 | 5, 6 | ax-mp 7 | . 2 |
8 | df-xr 7064 | . 2 | |
9 | 7, 8 | eleqtrri 2113 | 1 |
Colors of variables: wff set class |
Syntax hints: wcel 1393 cvv 2557 cun 2915 cpw 3359 cpr 3376 cr 6888 cpnf 7057 cmnf 7058 cxr 7059 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-un 4170 ax-cnex 6975 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-uni 3581 df-pnf 7062 df-mnf 7063 df-xr 7064 |
This theorem is referenced by: elxr 8696 xrltnr 8701 mnflt 8704 mnfltpnf 8706 nltmnf 8709 mnfle 8713 xrltnsym 8714 xrlttri3 8718 ngtmnft 8731 xrrebnd 8732 xrre2 8734 xrre3 8735 ge0gtmnf 8736 xnegcl 8745 xltnegi 8748 xrex 8756 elioc2 8805 elico2 8806 elicc2 8807 ioomax 8817 iccmax 8818 elioomnf 8837 unirnioo 8842 |
Copyright terms: Public domain | W3C validator |