Metamath Proof Explorer


Theorem eliccioo

Description: Membership in a closed interval of extended reals versus the same open interval. (Contributed by Thierry Arnoux, 18-Dec-2016)

Ref Expression
Assertion eliccioo ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵 ) → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝐶 = 𝐴𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ 𝐶 = 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 prunioo ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵 ) → ( ( 𝐴 (,) 𝐵 ) ∪ { 𝐴 , 𝐵 } ) = ( 𝐴 [,] 𝐵 ) )
2 1 eleq2d ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵 ) → ( 𝐶 ∈ ( ( 𝐴 (,) 𝐵 ) ∪ { 𝐴 , 𝐵 } ) ↔ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) )
3 2 biimpar ( ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵 ) ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐶 ∈ ( ( 𝐴 (,) 𝐵 ) ∪ { 𝐴 , 𝐵 } ) )
4 elun ( 𝐶 ∈ ( ( 𝐴 (,) 𝐵 ) ∪ { 𝐴 , 𝐵 } ) ↔ ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ 𝐶 ∈ { 𝐴 , 𝐵 } ) )
5 elprg ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) → ( 𝐶 ∈ { 𝐴 , 𝐵 } ↔ ( 𝐶 = 𝐴𝐶 = 𝐵 ) ) )
6 5 orbi2d ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) → ( ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ 𝐶 ∈ { 𝐴 , 𝐵 } ) ↔ ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ ( 𝐶 = 𝐴𝐶 = 𝐵 ) ) ) )
7 4 6 syl5bb ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) → ( 𝐶 ∈ ( ( 𝐴 (,) 𝐵 ) ∪ { 𝐴 , 𝐵 } ) ↔ ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ ( 𝐶 = 𝐴𝐶 = 𝐵 ) ) ) )
8 7 adantl ( ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵 ) ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐶 ∈ ( ( 𝐴 (,) 𝐵 ) ∪ { 𝐴 , 𝐵 } ) ↔ ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ ( 𝐶 = 𝐴𝐶 = 𝐵 ) ) ) )
9 3 8 mpbid ( ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵 ) ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ ( 𝐶 = 𝐴𝐶 = 𝐵 ) ) )
10 3orass ( ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ 𝐶 = 𝐴𝐶 = 𝐵 ) ↔ ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ ( 𝐶 = 𝐴𝐶 = 𝐵 ) ) )
11 3orcoma ( ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ 𝐶 = 𝐴𝐶 = 𝐵 ) ↔ ( 𝐶 = 𝐴𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ 𝐶 = 𝐵 ) )
12 10 11 bitr3i ( ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ ( 𝐶 = 𝐴𝐶 = 𝐵 ) ) ↔ ( 𝐶 = 𝐴𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ 𝐶 = 𝐵 ) )
13 9 12 sylib ( ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵 ) ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐶 = 𝐴𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ 𝐶 = 𝐵 ) )
14 lbicc2 ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵 ) → 𝐴 ∈ ( 𝐴 [,] 𝐵 ) )
15 14 adantr ( ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵 ) ∧ 𝐶 = 𝐴 ) → 𝐴 ∈ ( 𝐴 [,] 𝐵 ) )
16 eleq1 ( 𝐶 = 𝐴 → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) )
17 16 adantl ( ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵 ) ∧ 𝐶 = 𝐴 ) → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) )
18 15 17 mpbird ( ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵 ) ∧ 𝐶 = 𝐴 ) → 𝐶 ∈ ( 𝐴 [,] 𝐵 ) )
19 ioossicc ( 𝐴 (,) 𝐵 ) ⊆ ( 𝐴 [,] 𝐵 )
20 19 sseli ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) → 𝐶 ∈ ( 𝐴 [,] 𝐵 ) )
21 20 adantl ( ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵 ) ∧ 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ) → 𝐶 ∈ ( 𝐴 [,] 𝐵 ) )
22 ubicc2 ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵 ) → 𝐵 ∈ ( 𝐴 [,] 𝐵 ) )
23 22 adantr ( ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵 ) ∧ 𝐶 = 𝐵 ) → 𝐵 ∈ ( 𝐴 [,] 𝐵 ) )
24 eleq1 ( 𝐶 = 𝐵 → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ 𝐵 ∈ ( 𝐴 [,] 𝐵 ) ) )
25 24 adantl ( ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵 ) ∧ 𝐶 = 𝐵 ) → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ 𝐵 ∈ ( 𝐴 [,] 𝐵 ) ) )
26 23 25 mpbird ( ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵 ) ∧ 𝐶 = 𝐵 ) → 𝐶 ∈ ( 𝐴 [,] 𝐵 ) )
27 18 21 26 3jaodan ( ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵 ) ∧ ( 𝐶 = 𝐴𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ 𝐶 = 𝐵 ) ) → 𝐶 ∈ ( 𝐴 [,] 𝐵 ) )
28 13 27 impbida ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵 ) → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝐶 = 𝐴𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ 𝐶 = 𝐵 ) ) )