Step |
Hyp |
Ref |
Expression |
1 |
|
df-xr |
⊢ ℝ* = ( ℝ ∪ { +∞ , -∞ } ) |
2 |
1
|
eleq2i |
⊢ ( 𝐴 ∈ ℝ* ↔ 𝐴 ∈ ( ℝ ∪ { +∞ , -∞ } ) ) |
3 |
|
elun |
⊢ ( 𝐴 ∈ ( ℝ ∪ { +∞ , -∞ } ) ↔ ( 𝐴 ∈ ℝ ∨ 𝐴 ∈ { +∞ , -∞ } ) ) |
4 |
|
pnfex |
⊢ +∞ ∈ V |
5 |
|
mnfxr |
⊢ -∞ ∈ ℝ* |
6 |
5
|
elexi |
⊢ -∞ ∈ V |
7 |
4 6
|
elpr2 |
⊢ ( 𝐴 ∈ { +∞ , -∞ } ↔ ( 𝐴 = +∞ ∨ 𝐴 = -∞ ) ) |
8 |
7
|
orbi2i |
⊢ ( ( 𝐴 ∈ ℝ ∨ 𝐴 ∈ { +∞ , -∞ } ) ↔ ( 𝐴 ∈ ℝ ∨ ( 𝐴 = +∞ ∨ 𝐴 = -∞ ) ) ) |
9 |
|
3orass |
⊢ ( ( 𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞ ) ↔ ( 𝐴 ∈ ℝ ∨ ( 𝐴 = +∞ ∨ 𝐴 = -∞ ) ) ) |
10 |
8 9
|
bitr4i |
⊢ ( ( 𝐴 ∈ ℝ ∨ 𝐴 ∈ { +∞ , -∞ } ) ↔ ( 𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞ ) ) |
11 |
2 3 10
|
3bitri |
⊢ ( 𝐴 ∈ ℝ* ↔ ( 𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞ ) ) |