mnfltpnf - Intuitionistic Logic Explorer

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Theorem mnfltpnf 9721

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 2165	. . . 4
2		eqid 2165	. . . 4
3		olc 701	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
4	1, 2, 3	mp2an 423	. . 3 $/\$ $/\$ $\/$ $/\$
5	4	orci 721	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
6		mnfxr 7955	. . 3
7		pnfxr 7951	. . 3
8		ltxr 9711	. . 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 1343

wcel 2136 class class class wbr 3982

cr 7752

cltrr 7757

cpnf 7930

cmnf 7931

cxr 7932

clt 7933

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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-cnex 7844

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ral 2449 df-rex 2450 df-v 2728 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-xp 4610 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938

This theorem is referenced by: mnfltxr 9722 xrlttr 9731 xrltso 9732 xrlttri3 9733 nltpnft 9750 npnflt 9751 ngtmnft 9753 nmnfgt 9754 xltnegi 9771 xposdif 9818