mnfltpnf - Intuitionistic Logic Explorer

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Theorem mnfltpnf 9798

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 2187	. . . 4
2		eqid 2187	. . . 4
3		olc 712	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
4	1, 2, 3	mp2an 426	. . 3 $/\$ $/\$ $\/$ $/\$
5	4	orci 732	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
6		mnfxr 8027	. . 3
7		pnfxr 8023	. . 3
8		ltxr 9788	. . 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 709

wceq 1363

wcel 2158 class class class wbr 4015

cr 7823

cltrr 7828

cpnf 8002

cmnf 8003

cxr 8004

clt 8005

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 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-cnex 7915

This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-xp 4644 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010

This theorem is referenced by: mnfltxr 9799 xrlttr 9808 xrltso 9809 xrlttri3 9810 nltpnft 9827 npnflt 9828 ngtmnft 9830 nmnfgt 9831 xltnegi 9848 xposdif 9895