mnflt - Intuitionistic Logic Explorer

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Theorem mnflt 9740

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 2170	. . . 4
2		olc 706	. . . 4 $/\$ $/\$ $\/$ $/\$
3	1, 2	mpan 422	. . 3 $/\$ $\/$ $/\$
4	3	olcd 729	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
5		mnfxr 7976	. . 3
6		rexr 7965	. . 3
7		ltxr 9732	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
8	5, 6, 7	sylancr 412	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
9	4, 8	mpbird 166	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $\/$ wo 703

wceq 1348

wcel 2141 class class class wbr 3989

cr 7773

cltrr 7778

cpnf 7951

cmnf 7952

cxr 7953

clt 7954

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-cnex 7865

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-xp 4617 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959

This theorem is referenced by: mnflt0 9741 mnfltxr 9743 xrlttr 9752 xrltso 9753 xrlttri3 9754 ngtmnft 9774 nmnfgt 9775 xrrebnd 9776 xrre3 9779 xltnegi 9792 xltadd1 9833 xposdif 9839 elico2 9894 elicc2 9895 ioomax 9905 elioomnf 9925 qbtwnxr 10214 tgioo 13340