mnflt - Intuitionistic Logic Explorer

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Theorem mnflt 9251

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 2088	. . . 4
2		olc 667	. . . 4 $/\$ $/\$ $\/$ $/\$
3	1, 2	mpan 415	. . 3 $/\$ $\/$ $/\$
4	3	olcd 688	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
5		mnfxr 7542	. . 3
6		rexr 7531	. . 3
7		ltxr 9244	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
8	5, 6, 7	sylancr 405	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
9	4, 8	mpbird 165	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 102

wb 103 $\/$ wo 664

wceq 1289

wcel 1438 class class class wbr 3845

cr 7347

cltrr 7352

cpnf 7517

cmnf 7518

cxr 7519

clt 7520

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-cnex 7434

This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-xp 4444 df-pnf 7522 df-mnf 7523 df-xr 7524 df-ltxr 7525

This theorem is referenced by: mnflt0 9252 mnfltxr 9254 xrlttr 9263 xrltso 9264 xrlttri3 9265 ngtmnft 9278 xrrebnd 9279 xrre3 9282 xltnegi 9295 elico2 9353 elicc2 9354 ioomax 9364 elioomnf 9384 qbtwnxr 9665