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 * B A +∞ B A < B

Proof

Step Hyp Ref Expression
1 pnfxr +∞ *
2 elioo2 A * +∞ * B A +∞ B A < B B < +∞
3 1 2 mpan2 A * B A +∞ B A < B B < +∞
4 df-3an B A < B B < +∞ B A < B B < +∞
5 ltpnf B B < +∞
6 5 adantr B A < B B < +∞
7 6 pm4.71i B A < B B A < B B < +∞
8 4 7 bitr4i B A < B B < +∞ B A < B
9 3 8 bitrdi A * B A +∞ B A < B