mnfltpnf - Intuitionistic Logic Explorer

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Theorem mnfltpnf 9993

Description: Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.)

**Assertion**
Ref	Expression
mnfltpnf

**Proof of Theorem mnfltpnf**
Step	Hyp	Ref	Expression
1		eqid 2229	. . . 4
2		eqid 2229	. . . 4
3		olc 716	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
4	1, 2, 3	mp2an 426	. . 3 $/\$ $/\$ $\/$ $/\$
5	4	orci 736	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
6		mnfxr 8214	. . 3
7		pnfxr 8210	. . 3
8		ltxr 9983	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
9	6, 7, 8	mp2an 426	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
10	5, 9	mpbir 146	1

Colors of variables: wff set class

Syntax hints: $/\$ wa 104

wb 105 $\/$ wo 713

wceq 1395

wcel 2200 class class class wbr 4083

cr 8009

cltrr 8014

cpnf 8189

cmnf 8190

cxr 8191

clt 8192

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 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-cnex 8101

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ral 2513 df-rex 2514 df-v 2801 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-br 4084 df-opab 4146 df-xp 4725 df-pnf 8194 df-mnf 8195 df-xr 8196 df-ltxr 8197

This theorem is referenced by: mnfltxr 9994 xrlttr 10003 xrltso 10004 xrlttri3 10005 nltpnft 10022 npnflt 10023 ngtmnft 10025 nmnfgt 10026 xltnegi 10043 xposdif 10090