elxr - Intuitionistic Logic Explorer

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Theorem elxr 9712

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 7937	. . 3
2	1	eleq2i 2233	. 2
3		elun 3263	. 2 $\/$
4		pnfex 7952	. . . . 5
5		mnfxr 7955	. . . . . 6
6	5	elexi 2738	. . . . 5
7	4, 6	elpr2 3598	. . . 4 $\/$
8	7	orbi2i 752	. . 3 $\/$ $\/$ $\/$
9		3orass 971	. . 3 $\/$ $\/$ $\/$ $\/$
10	8, 9	bitr4i 186	. 2 $\/$ $\/$ $\/$
11	2, 3, 10	3bitri 205	1 $\/$ $\/$

Colors of variables: wff set class

Syntax hints:

wb 104 $\/$ wo 698 $\/$ w3o 967

wceq 1343

wcel 2136

cun 3114

cpr 3577

cr 7752

cpnf 7930

cmnf 7931

cxr 7932

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 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-un 4411 ax-cnex 7844

This theorem depends on definitions: df-bi 116 df-3or 969 df-tru 1346 df-nf 1449 df-sb 1751 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-rex 2450 df-v 2728 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-uni 3790 df-pnf 7935 df-mnf 7936 df-xr 7937

This theorem is referenced by: xrnemnf 9713 xrnepnf 9714 xrltnr 9715 xrltnsym 9729 xrlttr 9731 xrltso 9732 xrlttri3 9733 nltpnft 9750 npnflt 9751 ngtmnft 9753 nmnfgt 9754 xrrebnd 9755 xnegcl 9768 xnegneg 9769 xltnegi 9771 xrpnfdc 9778 xrmnfdc 9779 xnegid 9795 xaddcom 9797 xaddid1 9798 xnegdi 9804 xleadd1a 9809 xltadd1 9812 xlt2add 9816 xsubge0 9817 xposdif 9818 xleaddadd 9823 qbtwnxr 10193 xrmaxiflemcl 11186 xrmaxifle 11187 xrmaxiflemab 11188 xrmaxiflemlub 11189 xrmaxltsup 11199 xrmaxadd 11202 xrbdtri 11217 isxmet2d 12988 blssioo 13185