elxr - Intuitionistic Logic Explorer

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Theorem elxr 8696

Description: Membership in the set of extended reals. (Contributed by NM, 14-Oct-2005.)

**Assertion**
Ref	Expression
elxr	$\/$ $\/$

**Proof of Theorem elxr**
Step	Hyp	Ref	Expression
1		df-xr 7064	. . 3
2	1	eleq2i 2104	. 2
3		elun 3084	. 2 $\/$
4		pnfex 8693	. . . . 5
5		mnfxr 8694	. . . . . 6
6	5	elexi 2567	. . . . 5
7	4, 6	elpr2 3397	. . . 4 $\/$
8	7	orbi2i 679	. . 3 $\/$ $\/$ $\/$
9		3orass 888	. . 3 $\/$ $\/$ $\/$ $\/$
10	8, 9	bitr4i 176	. 2 $\/$ $\/$ $\/$
11	2, 3, 10	3bitri 195	1 $\/$ $\/$

Colors of variables: wff set class

Syntax hints:

wb 98 $\/$ wo 629 $\/$ w3o 884

wceq 1243

wcel 1393

cun 2915

cpr 3376

cr 6888

cpnf 7057

cmnf 7058

cxr 7059

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-un 4170 ax-cnex 6975

This theorem depends on definitions: df-bi 110 df-3or 886 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-uni 3581 df-pnf 7062 df-mnf 7063 df-xr 7064

This theorem is referenced by: xrnemnf 8699 xrnepnf 8700 xrltnr 8701 xrltnsym 8714 xrlttr 8716 xrltso 8717 xrlttri3 8718 nltpnft 8730 ngtmnft 8731 xrrebnd 8732 xnegcl 8745 xnegneg 8746 xltnegi 8748