elxr - Intuitionistic Logic Explorer

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Theorem elxr 9733

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 7958	. . 3
2	1	eleq2i 2237	. 2
3		elun 3268	. 2 $\/$
4		pnfex 7973	. . . . 5
5		mnfxr 7976	. . . . . 6
6	5	elexi 2742	. . . . 5
7	4, 6	elpr2 3605	. . . 4 $\/$
8	7	orbi2i 757	. . 3 $\/$ $\/$ $\/$
9		3orass 976	. . 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 703 $\/$ w3o 972

wceq 1348

wcel 2141

cun 3119

cpr 3584

cr 7773

cpnf 7951

cmnf 7952

cxr 7953

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 4107 ax-pow 4160 ax-un 4418 ax-cnex 7865

This theorem depends on definitions: df-bi 116 df-3or 974 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-uni 3797 df-pnf 7956 df-mnf 7957 df-xr 7958

This theorem is referenced by: xrnemnf 9734 xrnepnf 9735 xrltnr 9736 xrltnsym 9750 xrlttr 9752 xrltso 9753 xrlttri3 9754 nltpnft 9771 npnflt 9772 ngtmnft 9774 nmnfgt 9775 xrrebnd 9776 xnegcl 9789 xnegneg 9790 xltnegi 9792 xrpnfdc 9799 xrmnfdc 9800 xnegid 9816 xaddcom 9818 xaddid1 9819 xnegdi 9825 xleadd1a 9830 xltadd1 9833 xlt2add 9837 xsubge0 9838 xposdif 9839 xleaddadd 9844 qbtwnxr 10214 xrmaxiflemcl 11208 xrmaxifle 11209 xrmaxiflemab 11210 xrmaxiflemlub 11211 xrmaxltsup 11221 xrmaxadd 11224 xrbdtri 11239 isxmet2d 13142 blssioo 13339