mnfltpnf - Intuitionistic Logic Explorer

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Theorem mnfltpnf 9601

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 2140	. . . 4
2		eqid 2140	. . . 4
3		olc 701	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
4	1, 2, 3	mp2an 423	. . 3 $/\$ $/\$ $\/$ $/\$
5	4	orci 721	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
6		mnfxr 7846	. . 3
7		pnfxr 7842	. . 3
8		ltxr 9592	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
9	6, 7, 8	mp2an 423	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
10	5, 9	mpbir 145	1

Colors of variables: wff set class

Syntax hints: $/\$ wa 103

wb 104 $\/$ wo 698

wceq 1332

wcel 1481 class class class wbr 3937

cr 7643

cltrr 7648

cpnf 7821

cmnf 7822

cxr 7823

clt 7824

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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-cnex 7735

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829

This theorem is referenced by: mnfltxr 9602 xrlttr 9611 xrltso 9612 xrlttri3 9613 nltpnft 9627 npnflt 9628 ngtmnft 9630 nmnfgt 9631 xltnegi 9648 xposdif 9695