mnflt - Intuitionistic Logic Explorer

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Theorem mnflt 9849

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 2193	. . . 4
2		olc 712	. . . 4 $/\$ $/\$ $\/$ $/\$
3	1, 2	mpan 424	. . 3 $/\$ $\/$ $/\$
4	3	olcd 735	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
5		mnfxr 8076	. . 3
6		rexr 8065	. . 3
7		ltxr 9841	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
8	5, 6, 7	sylancr 414	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
9	4, 8	mpbird 167	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $\/$ wo 709

wceq 1364

wcel 2164 class class class wbr 4029

cr 7871

cltrr 7876

cpnf 8051

cmnf 8052

cxr 8053

clt 8054

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-cnex 7963

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-opab 4091 df-xp 4665 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059

This theorem is referenced by: mnflt0 9850 mnfltxr 9852 xrlttr 9861 xrltso 9862 xrlttri3 9863 ngtmnft 9883 nmnfgt 9884 xrrebnd 9885 xrre3 9888 xltnegi 9901 xltadd1 9942 xposdif 9948 elico2 10003 elicc2 10004 ioomax 10014 elioomnf 10034 qbtwnxr 10326 tgioo 14714