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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 2117 | . . . 4 | |
2 | olc 685 | . . . 4 | |
3 | 1, 2 | mpan 420 | . . 3 |
4 | 3 | olcd 708 | . 2 |
5 | mnfxr 7790 | . . 3 | |
6 | rexr 7779 | . . 3 | |
7 | ltxr 9517 | . . 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 682 wceq 1316 wcel 1465 class class class wbr 3899 cr 7587 cltrr 7592 cpnf 7765 cmnf 7766 cxr 7767 clt 7768 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-cnex 7679 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 |
This theorem is referenced by: mnflt0 9525 mnfltxr 9527 xrlttr 9536 xrltso 9537 xrlttri3 9538 ngtmnft 9555 nmnfgt 9556 xrrebnd 9557 xrre3 9560 xltnegi 9573 xltadd1 9614 xposdif 9620 elico2 9675 elicc2 9676 ioomax 9686 elioomnf 9706 qbtwnxr 9990 tgioo 12626 |
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