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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 2137 | . . . 4 | |
2 | olc 700 | . . . 4 | |
3 | 1, 2 | mpan 420 | . . 3 |
4 | 3 | olcd 723 | . 2 |
5 | mnfxr 7815 | . . 3 | |
6 | rexr 7804 | . . 3 | |
7 | ltxr 9555 | . . 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 3924 cr 7612 cltrr 7617 cpnf 7790 cmnf 7791 cxr 7792 clt 7793 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-cnex 7704 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 |
This theorem is referenced by: mnflt0 9563 mnfltxr 9565 xrlttr 9574 xrltso 9575 xrlttri3 9576 ngtmnft 9593 nmnfgt 9594 xrrebnd 9595 xrre3 9598 xltnegi 9611 xltadd1 9652 xposdif 9658 elico2 9713 elicc2 9714 ioomax 9724 elioomnf 9744 qbtwnxr 10028 tgioo 12704 |
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