mnfltpnf - Intuitionistic Logic Explorer

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Theorem mnfltpnf 9729

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 2170	. . . 4
2		eqid 2170	. . . 4
3		olc 706	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
4	1, 2, 3	mp2an 424	. . 3 $/\$ $/\$ $\/$ $/\$
5	4	orci 726	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
6		mnfxr 7963	. . 3
7		pnfxr 7959	. . 3
8		ltxr 9719	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
9	6, 7, 8	mp2an 424	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
10	5, 9	mpbir 145	1

Colors of variables: wff set class

Syntax hints: $/\$ wa 103

wb 104 $\/$ wo 703

wceq 1348

wcel 2141 class class class wbr 3987

cr 7760

cltrr 7765

cpnf 7938

cmnf 7939

cxr 7940

clt 7941

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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-cnex 7852

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-xp 4615 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946

This theorem is referenced by: mnfltxr 9730 xrlttr 9739 xrltso 9740 xrlttri3 9741 nltpnft 9758 npnflt 9759 ngtmnft 9761 nmnfgt 9762 xltnegi 9779 xposdif 9826