mnflt - Intuitionistic Logic Explorer

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Theorem mnflt 10079

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 2231	. . . 4
2		olc 719	. . . 4 $/\$ $/\$ $\/$ $/\$
3	1, 2	mpan 424	. . 3 $/\$ $\/$ $/\$
4	3	olcd 742	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
5		mnfxr 8295	. . 3
6		rexr 8284	. . 3
7		ltxr 10071	. . 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 716

wceq 1398

wcel 2202 class class class wbr 4093

cr 8091

cltrr 8096

cpnf 8270

cmnf 8271

cxr 8272

clt 8273

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-cnex 8183

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ral 2516 df-rex 2517 df-v 2805 df-un 3205 df-in 3207 df-ss 3214 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-br 4094 df-opab 4156 df-xp 4737 df-pnf 8275 df-mnf 8276 df-xr 8277 df-ltxr 8278

This theorem is referenced by: mnflt0 10080 mnfltxr 10082 xrlttr 10091 xrltso 10092 xrlttri3 10093 ngtmnft 10113 nmnfgt 10114 xrrebnd 10115 xrre3 10118 xltnegi 10131 xltadd1 10172 xposdif 10178 elico2 10233 elicc2 10234 ioomax 10244 elioomnf 10264 qbtwnxr 10580 tgioo 15365 repiecelem 16757 repiecele0 16758