mnflt - Intuitionistic Logic Explorer

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Theorem mnflt 9599

Description: Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.)

**Assertion**
Ref	Expression
mnflt

**Proof of Theorem mnflt**
Step	Hyp	Ref	Expression
1		eqid 2140	. . . 4
2		olc 701	. . . 4 $/\$ $/\$ $\/$ $/\$
3	1, 2	mpan 421	. . 3 $/\$ $\/$ $/\$
4	3	olcd 724	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
5		mnfxr 7846	. . 3
6		rexr 7835	. . 3
7		ltxr 9592	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
8	5, 6, 7	sylancr 411	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
9	4, 8	mpbird 166	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $\/$ wo 698

wceq 1332

wcel 1481 class class class wbr 3937

cr 7643

cltrr 7648

cpnf 7821

cmnf 7822

cxr 7823

clt 7824

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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-cnex 7735

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829

This theorem is referenced by: mnflt0 9600 mnfltxr 9602 xrlttr 9611 xrltso 9612 xrlttri3 9613 ngtmnft 9630 nmnfgt 9631 xrrebnd 9632 xrre3 9635 xltnegi 9648 xltadd1 9689 xposdif 9695 elico2 9750 elicc2 9751 ioomax 9761 elioomnf 9781 qbtwnxr 10066 tgioo 12754