mnfltpnf - Intuitionistic Logic Explorer

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Theorem mnfltpnf 9851

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 2193	. . . 4
2		eqid 2193	. . . 4
3		olc 712	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
4	1, 2, 3	mp2an 426	. . 3 $/\$ $/\$ $\/$ $/\$
5	4	orci 732	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
6		mnfxr 8076	. . 3
7		pnfxr 8072	. . 3
8		ltxr 9841	. . 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 1364

wcel 2164 class class class wbr 4029

cr 7871

cltrr 7876

cpnf 8051

cmnf 8052

cxr 8053

clt 8054

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-cnex 7963

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-opab 4091 df-xp 4665 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059

This theorem is referenced by: mnfltxr 9852 xrlttr 9861 xrltso 9862 xrlttri3 9863 nltpnft 9880 npnflt 9881 ngtmnft 9883 nmnfgt 9884 xltnegi 9901 xposdif 9948