Metamath Proof Explorer

Theorem elicopnf

Description: Membership in a closed unbounded interval of reals. (Contributed by Mario Carneiro, 16-Sep-2014)

Ref Expression
Assertion elicopnf 
|- ( A e. RR -> ( B e. ( A [,) +oo ) <-> ( B e. RR /\ A <_ B ) ) )

Proof

Step Hyp Ref Expression
1 pnfxr 
 |-  +oo e. RR*
2 elico2 
 |-  ( ( A e. RR /\ +oo e. RR* ) -> ( B e. ( A [,) +oo ) <-> ( B e. RR /\ A <_ B /\ B < +oo ) ) )
3 1 2 mpan2 
 |-  ( A e. RR -> ( B e. ( A [,) +oo ) <-> ( B e. RR /\ A <_ B /\ B < +oo ) ) )
4 ltpnf 
 |-  ( B e. RR -> B < +oo )
5 4 adantr 
 |-  ( ( B e. RR /\ A <_ B ) -> B < +oo )
6 5 pm4.71i 
 |-  ( ( B e. RR /\ A <_ B ) <-> ( ( B e. RR /\ A <_ B ) /\ B < +oo ) )
7 df-3an 
 |-  ( ( B e. RR /\ A <_ B /\ B < +oo ) <-> ( ( B e. RR /\ A <_ B ) /\ B < +oo ) )
8 6 7 bitr4i 
 |-  ( ( B e. RR /\ A <_ B ) <-> ( B e. RR /\ A <_ B /\ B < +oo ) )
9 3 8 bitr4di 
 |-  ( A e. RR -> ( B e. ( A [,) +oo ) <-> ( B e. RR /\ A <_ B ) ) )