mnfltpnf - Intuitionistic Logic Explorer

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Theorem mnfltpnf 10064

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 2231	. . . 4
2		eqid 2231	. . . 4
3		olc 719	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
4	1, 2, 3	mp2an 426	. . 3 $/\$ $/\$ $\/$ $/\$
5	4	orci 739	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
6		mnfxr 8278	. . 3
7		pnfxr 8274	. . 3
8		ltxr 10054	. . 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 716

wceq 1398

wcel 2202 class class class wbr 4093

cr 8074

cltrr 8079

cpnf 8253

cmnf 8254

cxr 8255

clt 8256

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-cnex 8166

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ral 2516 df-rex 2517 df-v 2805 df-un 3205 df-in 3207 df-ss 3214 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-br 4094 df-opab 4156 df-xp 4737 df-pnf 8258 df-mnf 8259 df-xr 8260 df-ltxr 8261

This theorem is referenced by: mnfltxr 10065 xrlttr 10074 xrltso 10075 xrlttri3 10076 nltpnft 10093 npnflt 10094 ngtmnft 10096 nmnfgt 10097 xltnegi 10114 xposdif 10161