Metamath Proof Explorer

Theorem xrdifh

Description: Class difference of a half-open interval in the extended reals. (Contributed by Thierry Arnoux, 1-Aug-2017)

Ref Expression
Hypothesis xrdifh.1 ⊢ 𝐴 ∈ ℝ*
Assertion xrdifh ( ℝ* ∖ ( 𝐴 [,] +∞ ) ) = ( -∞ [,) 𝐴 )

Proof

Step Hyp Ref Expression
1 xrdifh.1 ⊢ 𝐴 ∈ ℝ*
2 biortn ⊢ ( 𝑥 ∈ ℝ* → ( ( ¬ 𝐴 ≤ 𝑥 ∨ ¬ 𝑥 ≤ +∞ ) ↔ ( ¬ 𝑥 ∈ ℝ* ∨ ( ¬ 𝐴 ≤ 𝑥 ∨ ¬ 𝑥 ≤ +∞ ) ) ) )
3 pnfge ⊢ ( 𝑥 ∈ ℝ* → 𝑥 ≤ +∞ )
4 3 notnotd ⊢ ( 𝑥 ∈ ℝ* → ¬ ¬ 𝑥 ≤ +∞ )
5 biorf ⊢ ( ¬ ¬ 𝑥 ≤ +∞ → ( ¬ 𝐴 ≤ 𝑥 ↔ ( ¬ 𝑥 ≤ +∞ ∨ ¬ 𝐴 ≤ 𝑥 ) ) )
6 4 5 syl ⊢ ( 𝑥 ∈ ℝ* → ( ¬ 𝐴 ≤ 𝑥 ↔ ( ¬ 𝑥 ≤ +∞ ∨ ¬ 𝐴 ≤ 𝑥 ) ) )
7 orcom ⊢ ( ( ¬ 𝐴 ≤ 𝑥 ∨ ¬ 𝑥 ≤ +∞ ) ↔ ( ¬ 𝑥 ≤ +∞ ∨ ¬ 𝐴 ≤ 𝑥 ) )
8 6 7 bitr4di ⊢ ( 𝑥 ∈ ℝ* → ( ¬ 𝐴 ≤ 𝑥 ↔ ( ¬ 𝐴 ≤ 𝑥 ∨ ¬ 𝑥 ≤ +∞ ) ) )
9 pnfxr ⊢ +∞ ∈ ℝ*
10 elicc1 ⊢ ( ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ) → ( 𝑥 ∈ ( 𝐴 [,] +∞ ) ↔ ( 𝑥 ∈ ℝ* ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ +∞ ) ) )
11 1 9 10 mp2an ⊢ ( 𝑥 ∈ ( 𝐴 [,] +∞ ) ↔ ( 𝑥 ∈ ℝ* ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ +∞ ) )
12 11 notbii ⊢ ( ¬ 𝑥 ∈ ( 𝐴 [,] +∞ ) ↔ ¬ ( 𝑥 ∈ ℝ* ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ +∞ ) )
13 3ianor ⊢ ( ¬ ( 𝑥 ∈ ℝ* ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ +∞ ) ↔ ( ¬ 𝑥 ∈ ℝ* ∨ ¬ 𝐴 ≤ 𝑥 ∨ ¬ 𝑥 ≤ +∞ ) )
14 3orass ⊢ ( ( ¬ 𝑥 ∈ ℝ* ∨ ¬ 𝐴 ≤ 𝑥 ∨ ¬ 𝑥 ≤ +∞ ) ↔ ( ¬ 𝑥 ∈ ℝ* ∨ ( ¬ 𝐴 ≤ 𝑥 ∨ ¬ 𝑥 ≤ +∞ ) ) )
15 12 13 14 3bitri ⊢ ( ¬ 𝑥 ∈ ( 𝐴 [,] +∞ ) ↔ ( ¬ 𝑥 ∈ ℝ* ∨ ( ¬ 𝐴 ≤ 𝑥 ∨ ¬ 𝑥 ≤ +∞ ) ) )
16 15 a1i ⊢ ( 𝑥 ∈ ℝ* → ( ¬ 𝑥 ∈ ( 𝐴 [,] +∞ ) ↔ ( ¬ 𝑥 ∈ ℝ* ∨ ( ¬ 𝐴 ≤ 𝑥 ∨ ¬ 𝑥 ≤ +∞ ) ) ) )
17 2 8 16 3bitr4rd ⊢ ( 𝑥 ∈ ℝ* → ( ¬ 𝑥 ∈ ( 𝐴 [,] +∞ ) ↔ ¬ 𝐴 ≤ 𝑥 ) )
18 xrltnle ⊢ ( ( 𝑥 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → ( 𝑥 < 𝐴 ↔ ¬ 𝐴 ≤ 𝑥 ) )
19 1 18 mpan2 ⊢ ( 𝑥 ∈ ℝ* → ( 𝑥 < 𝐴 ↔ ¬ 𝐴 ≤ 𝑥 ) )
20 17 19 bitr4d ⊢ ( 𝑥 ∈ ℝ* → ( ¬ 𝑥 ∈ ( 𝐴 [,] +∞ ) ↔ 𝑥 < 𝐴 ) )
21 20 pm5.32i ⊢ ( ( 𝑥 ∈ ℝ* ∧ ¬ 𝑥 ∈ ( 𝐴 [,] +∞ ) ) ↔ ( 𝑥 ∈ ℝ* ∧ 𝑥 < 𝐴 ) )
22 eldif ⊢ ( 𝑥 ∈ ( ℝ* ∖ ( 𝐴 [,] +∞ ) ) ↔ ( 𝑥 ∈ ℝ* ∧ ¬ 𝑥 ∈ ( 𝐴 [,] +∞ ) ) )
23 3anass ⊢ ( ( 𝑥 ∈ ℝ* ∧ -∞ ≤ 𝑥 ∧ 𝑥 < 𝐴 ) ↔ ( 𝑥 ∈ ℝ* ∧ ( -∞ ≤ 𝑥 ∧ 𝑥 < 𝐴 ) ) )
24 mnfxr ⊢ -∞ ∈ ℝ*
25 elico1 ⊢ ( ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → ( 𝑥 ∈ ( -∞ [,) 𝐴 ) ↔ ( 𝑥 ∈ ℝ* ∧ -∞ ≤ 𝑥 ∧ 𝑥 < 𝐴 ) ) )
26 24 1 25 mp2an ⊢ ( 𝑥 ∈ ( -∞ [,) 𝐴 ) ↔ ( 𝑥 ∈ ℝ* ∧ -∞ ≤ 𝑥 ∧ 𝑥 < 𝐴 ) )
27 mnfle ⊢ ( 𝑥 ∈ ℝ* → -∞ ≤ 𝑥 )
28 27 biantrurd ⊢ ( 𝑥 ∈ ℝ* → ( 𝑥 < 𝐴 ↔ ( -∞ ≤ 𝑥 ∧ 𝑥 < 𝐴 ) ) )
29 28 pm5.32i ⊢ ( ( 𝑥 ∈ ℝ* ∧ 𝑥 < 𝐴 ) ↔ ( 𝑥 ∈ ℝ* ∧ ( -∞ ≤ 𝑥 ∧ 𝑥 < 𝐴 ) ) )
30 23 26 29 3bitr4i ⊢ ( 𝑥 ∈ ( -∞ [,) 𝐴 ) ↔ ( 𝑥 ∈ ℝ* ∧ 𝑥 < 𝐴 ) )
31 21 22 30 3bitr4i ⊢ ( 𝑥 ∈ ( ℝ* ∖ ( 𝐴 [,] +∞ ) ) ↔ 𝑥 ∈ ( -∞ [,) 𝐴 ) )
32 31 eqriv ⊢ ( ℝ* ∖ ( 𝐴 [,] +∞ ) ) = ( -∞ [,) 𝐴 )