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Metamath Proof Explorer

Theorem elxr

Description: Membership in the set of extended reals. (Contributed by NM, 14-Oct-2005)

Ref Expression
Assertion elxr 
|- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )

Proof

Step Hyp Ref Expression
1 df-xr 
 |-  RR* = ( RR u. { +oo , -oo } )
2 1 eleq2i 
 |-  ( A e. RR* <-> A e. ( RR u. { +oo , -oo } ) )
3 elun 
 |-  ( A e. ( RR u. { +oo , -oo } ) <-> ( A e. RR \/ A e. { +oo , -oo } ) )
4 pnfex 
 |-  +oo e. _V
5 mnfxr 
 |-  -oo e. RR*
6 5 elexi 
 |-  -oo e. _V
7 4 6 elpr2 
 |-  ( A e. { +oo , -oo } <-> ( A = +oo \/ A = -oo ) )
8 7 orbi2i 
 |-  ( ( A e. RR \/ A e. { +oo , -oo } ) <-> ( A e. RR \/ ( A = +oo \/ A = -oo ) ) )
9 3orass 
 |-  ( ( A e. RR \/ A = +oo \/ A = -oo ) <-> ( A e. RR \/ ( A = +oo \/ A = -oo ) ) )
10 8 9 bitr4i 
 |-  ( ( A e. RR \/ A e. { +oo , -oo } ) <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
11 2 3 10 3bitri 
 |-  ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )