elxr - Intuitionistic Logic Explorer

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Theorem elxr 9776

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 7996	. . 3
2	1	eleq2i 2244	. 2
3		elun 3277	. 2 $\/$
4		pnfex 8011	. . . . 5
5		mnfxr 8014	. . . . . 6
6	5	elexi 2750	. . . . 5
7	4, 6	elpr2 3615	. . . 4 $\/$
8	7	orbi2i 762	. . 3 $\/$ $\/$ $\/$
9		3orass 981	. . 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 708 $\/$ w3o 977

wceq 1353

wcel 2148

cun 3128

cpr 3594

cr 7810

cpnf 7989

cmnf 7990

cxr 7991

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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-un 4434 ax-cnex 7902

This theorem depends on definitions: df-bi 117 df-3or 979 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-v 2740 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-uni 3811 df-pnf 7994 df-mnf 7995 df-xr 7996

This theorem is referenced by: xrnemnf 9777 xrnepnf 9778 xrltnr 9779 xrltnsym 9793 xrlttr 9795 xrltso 9796 xrlttri3 9797 nltpnft 9814 npnflt 9815 ngtmnft 9817 nmnfgt 9818 xrrebnd 9819 xnegcl 9832 xnegneg 9833 xltnegi 9835 xrpnfdc 9842 xrmnfdc 9843 xnegid 9859 xaddcom 9861 xaddid1 9862 xnegdi 9868 xleadd1a 9873 xltadd1 9876 xlt2add 9880 xsubge0 9881 xposdif 9882 xleaddadd 9887 qbtwnxr 10258 xrmaxiflemcl 11253 xrmaxifle 11254 xrmaxiflemab 11255 xrmaxiflemlub 11256 xrmaxltsup 11266 xrmaxadd 11269 xrbdtri 11284 isxmet2d 13851 blssioo 14048