mnflt - Intuitionistic Logic Explorer

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Theorem mnflt 10202

Description: Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.)

**Assertion**
Ref	Expression
mnflt

**Proof of Theorem mnflt**
Step	Hyp	Ref	Expression
1		eqid 2238	. . . 4
2		olc 723	. . . 4 $/\$ $/\$ $\/$ $/\$
3	1, 2	mpan 428	. . 3 $/\$ $\/$ $/\$
4	3	olcd 746	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
5		mnfxr 8388	. . 3
6		rexr 8377	. . 3
7		ltxr 10194	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
8	5, 6, 7	sylancr 418	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
9	4, 8	mpbird 167	1

Colors of variables: wff set class

This proof depends on syntax axioms:

wi 4 $/\$ wa 104

wb 105 $\/$ wo 720

wceq 1402

wcel 2209 class class class wbr 4131

cr 8184

cltrr 8189

cpnf 8363

cmnf 8364

cxr 8365

clt 8366

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4251 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-cnex 8276

This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ral 2533 df-rex 2534 df-v 2823 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-br 4132 df-opab 4194 df-xp 4782 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371

This theorem is used by: mnflt0 10203 mnfltxr 10205 xrlttr 10214 xrltso 10215 xrlttri3 10216 ngtmnft 10236 nmnfgt 10237 xrrebnd 10238 xrre3 10241 xltnegi 10254 xltadd1 10295 xposdif 10301 elico2 10356 elicc2 10357 ioomax 10367 elioomnf 10387 qbtwnxr 10709 tgioo 15780 repiecelem 17280 repiecele0 17281