mnflt - Intuitionistic Logic Explorer

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Theorem mnflt 9852

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 8078	. . 3
6		rexr 8067	. . 3
7		ltxr 9844	. . 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 4030

cr 7873

cltrr 7878

cpnf 8053

cmnf 8054

cxr 8055

clt 8056

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 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-cnex 7965

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 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-br 4031 df-opab 4092 df-xp 4666 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061

This theorem is referenced by: mnflt0 9853 mnfltxr 9855 xrlttr 9864 xrltso 9865 xrlttri3 9866 ngtmnft 9886 nmnfgt 9887 xrrebnd 9888 xrre3 9891 xltnegi 9904 xltadd1 9945 xposdif 9951 elico2 10006 elicc2 10007 ioomax 10017 elioomnf 10037 qbtwnxr 10329 tgioo 14733