Metamath Proof Explorer

Theorem elioopnf

Description: Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014)

Ref Expression
Assertion elioopnf 
|- ( A e. RR* -> ( B e. ( A (,) +oo ) <-> ( B e. RR /\ A < B ) ) )

Proof

Step Hyp Ref Expression
1 pnfxr 
 |-  +oo e. RR*
2 elioo2 
 |-  ( ( 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 df-3an 
 |-  ( ( B e. RR /\ A < B /\ B < +oo ) <-> ( ( B e. RR /\ A < B ) /\ B < +oo ) )
5 ltpnf 
 |-  ( B e. RR -> B < +oo )
6 5 adantr 
 |-  ( ( B e. RR /\ A < B ) -> B < +oo )
7 6 pm4.71i 
 |-  ( ( B e. RR /\ A < B ) <-> ( ( B e. RR /\ A < B ) /\ B < +oo ) )
8 4 7 bitr4i 
 |-  ( ( B e. RR /\ A < B /\ B < +oo ) <-> ( B e. RR /\ A < B ) )
9 3 8 bitrdi 
 |-  ( A e. RR* -> ( B e. ( A (,) +oo ) <-> ( B e. RR /\ A < B ) ) )