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Mirrors > Home > ILE Home > Th. List > mnflt | Unicode version |
Description: Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.) |
Ref | Expression |
---|---|
mnflt |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2139 | . . . 4 | |
2 | olc 700 | . . . 4 | |
3 | 1, 2 | mpan 420 | . . 3 |
4 | 3 | olcd 723 | . 2 |
5 | mnfxr 7822 | . . 3 | |
6 | rexr 7811 | . . 3 | |
7 | ltxr 9562 | . . 3 | |
8 | 5, 6, 7 | sylancr 410 | . 2 |
9 | 4, 8 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 697 wceq 1331 wcel 1480 class class class wbr 3929 cr 7619 cltrr 7624 cpnf 7797 cmnf 7798 cxr 7799 clt 7800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-cnex 7711 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 |
This theorem is referenced by: mnflt0 9570 mnfltxr 9572 xrlttr 9581 xrltso 9582 xrlttri3 9583 ngtmnft 9600 nmnfgt 9601 xrrebnd 9602 xrre3 9605 xltnegi 9618 xltadd1 9659 xposdif 9665 elico2 9720 elicc2 9721 ioomax 9731 elioomnf 9751 qbtwnxr 10035 tgioo 12715 |
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