elxr - Intuitionistic Logic Explorer

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Theorem elxr 8850

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 7157	. . 3
2	1	eleq2i 2145	. 2
3		elun 3113	. 2 $\/$
4		pnfex 8847	. . . . 5
5		mnfxr 8848	. . . . . 6
6	5	elexi 2611	. . . . 5
7	4, 6	elpr2 3420	. . . 4 $\/$
8	7	orbi2i 711	. . 3 $\/$ $\/$ $\/$
9		3orass 922	. . 3 $\/$ $\/$ $\/$ $\/$
10	8, 9	bitr4i 185	. 2 $\/$ $\/$ $\/$
11	2, 3, 10	3bitri 204	1 $\/$ $\/$

Colors of variables: wff set class

Syntax hints:

wb 103 $\/$ wo 661 $\/$ w3o 918

wceq 1284

wcel 1433

cun 2971

cpr 3399

cr 6980

cpnf 7150

cmnf 7151

cxr 7152

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-un 4188 ax-cnex 7067

This theorem depends on definitions: df-bi 115 df-3or 920 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-uni 3602 df-pnf 7155 df-mnf 7156 df-xr 7157

This theorem is referenced by: xrnemnf 8853 xrnepnf 8854 xrltnr 8855 xrltnsym 8868 xrlttr 8870 xrltso 8871 xrlttri3 8872 nltpnft 8884 ngtmnft 8885 xrrebnd 8886 xnegcl 8899 xnegneg 8900 xltnegi 8902 qbtwnxr 9266