mnflt - Intuitionistic Logic Explorer

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Theorem mnflt 9719

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 2165	. . . 4
2		olc 701	. . . 4 $/\$ $/\$ $\/$ $/\$
3	1, 2	mpan 421	. . 3 $/\$ $\/$ $/\$
4	3	olcd 724	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
5		mnfxr 7955	. . 3
6		rexr 7944	. . 3
7		ltxr 9711	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
8	5, 6, 7	sylancr 411	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
9	4, 8	mpbird 166	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $\/$ wo 698

wceq 1343

wcel 2136 class class class wbr 3982

cr 7752

cltrr 7757

cpnf 7930

cmnf 7931

cxr 7932

clt 7933

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 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-cnex 7844

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ral 2449 df-rex 2450 df-v 2728 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-xp 4610 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938

This theorem is referenced by: mnflt0 9720 mnfltxr 9722 xrlttr 9731 xrltso 9732 xrlttri3 9733 ngtmnft 9753 nmnfgt 9754 xrrebnd 9755 xrre3 9758 xltnegi 9771 xltadd1 9812 xposdif 9818 elico2 9873 elicc2 9874 ioomax 9884 elioomnf 9904 qbtwnxr 10193 tgioo 13186