mnflt - Intuitionistic Logic Explorer

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Theorem mnflt 9940

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 2207	. . . 4
2		olc 713	. . . 4 $/\$ $/\$ $\/$ $/\$
3	1, 2	mpan 424	. . 3 $/\$ $\/$ $/\$
4	3	olcd 736	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
5		mnfxr 8164	. . 3
6		rexr 8153	. . 3
7		ltxr 9932	. . 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 710

wceq 1373

wcel 2178 class class class wbr 4059

cr 7959

cltrr 7964

cpnf 8139

cmnf 8140

cxr 8141

clt 8142

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-sep 4178 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-cnex 8051

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ral 2491 df-rex 2492 df-v 2778 df-un 3178 df-in 3180 df-ss 3187 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-br 4060 df-opab 4122 df-xp 4699 df-pnf 8144 df-mnf 8145 df-xr 8146 df-ltxr 8147

This theorem is referenced by: mnflt0 9941 mnfltxr 9943 xrlttr 9952 xrltso 9953 xrlttri3 9954 ngtmnft 9974 nmnfgt 9975 xrrebnd 9976 xrre3 9979 xltnegi 9992 xltadd1 10033 xposdif 10039 elico2 10094 elicc2 10095 ioomax 10105 elioomnf 10125 qbtwnxr 10437 tgioo 15141