mnflt - Intuitionistic Logic Explorer

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Theorem mnflt 8704

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 2040	. . . 4
2		olc 632	. . . 4 $/\$ $/\$ $\/$ $/\$
3	1, 2	mpan 400	. . 3 $/\$ $\/$ $/\$
4	3	olcd 653	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
5		mnfxr 8694	. . 3
6		rexr 7071	. . 3
7		ltxr 8695	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
8	5, 6, 7	sylancr 393	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
9	4, 8	mpbird 156	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 97

wb 98 $\/$ wo 629

wceq 1243

wcel 1393 class class class wbr 3764

cr 6888

cltrr 6893

cpnf 7057

cmnf 7058

cxr 7059

clt 7060

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-cnex 6975

This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065

This theorem is referenced by: mnflt0 8705 mnfltxr 8707 xrlttr 8716 xrltso 8717 xrlttri3 8718 ngtmnft 8731 xrrebnd 8732 xrre3 8735 xltnegi 8748 elico2 8806 elicc2 8807 ioomax 8817 elioomnf 8837