elxr - Intuitionistic Logic Explorer

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Theorem elxr 9845

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 8060	. . 3
2	1	eleq2i 2260	. 2
3		elun 3301	. 2 $\/$
4		pnfex 8075	. . . . 5
5		mnfxr 8078	. . . . . 6
6	5	elexi 2772	. . . . 5
7	4, 6	elpr2 3641	. . . 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 3152

cpr 3620

cr 7873

cpnf 8053

cmnf 8054

cxr 8055

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 4148 ax-pow 4204 ax-un 4465 ax-cnex 7965

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 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-uni 3837 df-pnf 8058 df-mnf 8059 df-xr 8060

This theorem is referenced by: xrnemnf 9846 xrnepnf 9847 xrltnr 9848 xrltnsym 9862 xrlttr 9864 xrltso 9865 xrlttri3 9866 nltpnft 9883 npnflt 9884 ngtmnft 9886 nmnfgt 9887 xrrebnd 9888 xnegcl 9901 xnegneg 9902 xltnegi 9904 xrpnfdc 9911 xrmnfdc 9912 xnegid 9928 xaddcom 9930 xaddid1 9931 xnegdi 9937 xleadd1a 9942 xltadd1 9945 xlt2add 9949 xsubge0 9950 xposdif 9951 xleaddadd 9956 qbtwnxr 10329 xrmaxiflemcl 11391 xrmaxifle 11392 xrmaxiflemab 11393 xrmaxiflemlub 11394 xrmaxltsup 11404 xrmaxadd 11407 xrbdtri 11422 isxmet2d 14527 blssioo 14732