elxr - Intuitionistic Logic Explorer

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Theorem elxr 9842

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 8058	. . 3
2	1	eleq2i 2260	. 2
3		elun 3300	. 2 $\/$
4		pnfex 8073	. . . . 5
5		mnfxr 8076	. . . . . 6
6	5	elexi 2772	. . . . 5
7	4, 6	elpr2 3640	. . . 4 $\/$
8	7	orbi2i 763	. . 3 $\/$ $\/$ $\/$
9		3orass 983	. . 3 $\/$ $\/$ $\/$ $\/$
10	8, 9	bitr4i 187	. 2 $\/$ $\/$ $\/$
11	2, 3, 10	3bitri 206	1 $\/$ $\/$

Colors of variables: wff set class

Syntax hints:

wb 105 $\/$ wo 709 $\/$ w3o 979

wceq 1364

wcel 2164

cun 3151

cpr 3619

cr 7871

cpnf 8051

cmnf 8052

cxr 8053

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-un 4464 ax-cnex 7963

This theorem depends on definitions: df-bi 117 df-3or 981 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-rex 2478 df-v 2762 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-uni 3836 df-pnf 8056 df-mnf 8057 df-xr 8058

This theorem is referenced by: xrnemnf 9843 xrnepnf 9844 xrltnr 9845 xrltnsym 9859 xrlttr 9861 xrltso 9862 xrlttri3 9863 nltpnft 9880 npnflt 9881 ngtmnft 9883 nmnfgt 9884 xrrebnd 9885 xnegcl 9898 xnegneg 9899 xltnegi 9901 xrpnfdc 9908 xrmnfdc 9909 xnegid 9925 xaddcom 9927 xaddid1 9928 xnegdi 9934 xleadd1a 9939 xltadd1 9942 xlt2add 9946 xsubge0 9947 xposdif 9948 xleaddadd 9953 qbtwnxr 10326 xrmaxiflemcl 11388 xrmaxifle 11389 xrmaxiflemab 11390 xrmaxiflemlub 11391 xrmaxltsup 11401 xrmaxadd 11404 xrbdtri 11419 isxmet2d 14516 blssioo 14713