# 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 ${⊢}{A}\in {ℝ}^{*}\to \left({B}\in \left({A},\mathrm{+\infty }\right)↔\left({B}\in ℝ\wedge {A}<{B}\right)\right)$

### Proof

Step Hyp Ref Expression
1 pnfxr ${⊢}\mathrm{+\infty }\in {ℝ}^{*}$
2 elioo2 ${⊢}\left({A}\in {ℝ}^{*}\wedge \mathrm{+\infty }\in {ℝ}^{*}\right)\to \left({B}\in \left({A},\mathrm{+\infty }\right)↔\left({B}\in ℝ\wedge {A}<{B}\wedge {B}<\mathrm{+\infty }\right)\right)$
3 1 2 mpan2 ${⊢}{A}\in {ℝ}^{*}\to \left({B}\in \left({A},\mathrm{+\infty }\right)↔\left({B}\in ℝ\wedge {A}<{B}\wedge {B}<\mathrm{+\infty }\right)\right)$
4 df-3an ${⊢}\left({B}\in ℝ\wedge {A}<{B}\wedge {B}<\mathrm{+\infty }\right)↔\left(\left({B}\in ℝ\wedge {A}<{B}\right)\wedge {B}<\mathrm{+\infty }\right)$
5 ltpnf ${⊢}{B}\in ℝ\to {B}<\mathrm{+\infty }$
6 5 adantr ${⊢}\left({B}\in ℝ\wedge {A}<{B}\right)\to {B}<\mathrm{+\infty }$
7 6 pm4.71i ${⊢}\left({B}\in ℝ\wedge {A}<{B}\right)↔\left(\left({B}\in ℝ\wedge {A}<{B}\right)\wedge {B}<\mathrm{+\infty }\right)$
8 4 7 bitr4i ${⊢}\left({B}\in ℝ\wedge {A}<{B}\wedge {B}<\mathrm{+\infty }\right)↔\left({B}\in ℝ\wedge {A}<{B}\right)$
9 3 8 syl6bb ${⊢}{A}\in {ℝ}^{*}\to \left({B}\in \left({A},\mathrm{+\infty }\right)↔\left({B}\in ℝ\wedge {A}<{B}\right)\right)$