mnfltpnf - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > mnfltpnf		Unicode version

Theorem mnfltpnf 9942

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 2207	. . . 4
2		eqid 2207	. . . 4
3		olc 713	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
4	1, 2, 3	mp2an 426	. . 3 $/\$ $/\$ $\/$ $/\$
5	4	orci 733	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
6		mnfxr 8164	. . 3
7		pnfxr 8160	. . 3
8		ltxr 9932	. . 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 710

wceq 1373

wcel 2178 class class class wbr 4059

cr 7959

cltrr 7964

cpnf 8139

cmnf 8140

cxr 8141

clt 8142

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-sep 4178 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-cnex 8051

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ral 2491 df-rex 2492 df-v 2778 df-un 3178 df-in 3180 df-ss 3187 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-br 4060 df-opab 4122 df-xp 4699 df-pnf 8144 df-mnf 8145 df-xr 8146 df-ltxr 8147

This theorem is referenced by: mnfltxr 9943 xrlttr 9952 xrltso 9953 xrlttri3 9954 nltpnft 9971 npnflt 9972 ngtmnft 9974 nmnfgt 9975 xltnegi 9992 xposdif 10039