mnfltpnf - Intuitionistic Logic Explorer

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Theorem mnfltpnf 8860

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 2081	. . . 4
2		eqid 2081	. . . 4
3		olc 664	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
4	1, 2, 3	mp2an 416	. . 3 $/\$ $/\$ $\/$ $/\$
5	4	orci 682	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
6		mnfxr 8848	. . 3
7		pnfxr 8846	. . 3
8		ltxr 8849	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
9	6, 7, 8	mp2an 416	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
10	5, 9	mpbir 144	1

Colors of variables: wff set class

Syntax hints: $/\$ wa 102

wb 103 $\/$ wo 661

wceq 1284

wcel 1433 class class class wbr 3785

cr 6980

cltrr 6985

cpnf 7150

cmnf 7151

cxr 7152

clt 7153

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 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-cnex 7067

This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158

This theorem is referenced by: mnfltxr 8861 xrlttr 8870 xrltso 8871 xrlttri3 8872 nltpnft 8884 ngtmnft 8885 xltnegi 8902