mnfltpnf - Intuitionistic Logic Explorer

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Theorem mnfltpnf 9354

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 2095	. . . 4
2		eqid 2095	. . . 4
3		olc 670	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
4	1, 2, 3	mp2an 418	. . 3 $/\$ $/\$ $\/$ $/\$
5	4	orci 688	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
6		mnfxr 7641	. . 3
7		pnfxr 7637	. . 3
8		ltxr 9345	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
9	6, 7, 8	mp2an 418	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
10	5, 9	mpbir 145	1

Colors of variables: wff set class

Syntax hints: $/\$ wa 103

wb 104 $\/$ wo 667

wceq 1296

wcel 1445 class class class wbr 3867

cr 7446

cltrr 7451

cpnf 7616

cmnf 7617

cxr 7618

clt 7619

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-cnex 7533

This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-xp 4473 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624

This theorem is referenced by: mnfltxr 9355 xrlttr 9364 xrltso 9365 xrlttri3 9366 nltpnft 9380 npnflt 9381 ngtmnft 9383 nmnfgt 9384 xltnegi 9401 xposdif 9448