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Metamath Proof Explorer

Theorem elicc2

Description: Membership in a closed real interval. (Contributed by Paul Chapman, 21-Sep-2007) (Revised by Mario Carneiro, 14-Jun-2014)

Ref Expression
Assertion elicc2 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 rexr ⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℝ* )
2 rexr ⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℝ* )
3 elicc1 ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) )
4 1 2 3 syl2an ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) )
5 mnfxr ⊢ -∞ ∈ ℝ*
6 5 a1i ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) → -∞ ∈ ℝ* )
7 1 ad2antrr ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) → 𝐴 ∈ ℝ* )
8 simpr1 ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) → 𝐶 ∈ ℝ* )
9 mnflt ⊢ ( 𝐴 ∈ ℝ → -∞ < 𝐴 )
10 9 ad2antrr ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) → -∞ < 𝐴 )
11 simpr2 ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) → 𝐴 ≤ 𝐶 )
12 6 7 8 10 11 xrltletrd ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) → -∞ < 𝐶 )
13 2 ad2antlr ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) → 𝐵 ∈ ℝ* )
14 pnfxr ⊢ +∞ ∈ ℝ*
15 14 a1i ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) → +∞ ∈ ℝ* )
16 simpr3 ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) → 𝐶 ≤ 𝐵 )
17 ltpnf ⊢ ( 𝐵 ∈ ℝ → 𝐵 < +∞ )
18 17 ad2antlr ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) → 𝐵 < +∞ )
19 8 13 15 16 18 xrlelttrd ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) → 𝐶 < +∞ )
20 xrrebnd ⊢ ( 𝐶 ∈ ℝ* → ( 𝐶 ∈ ℝ ↔ ( -∞ < 𝐶 ∧ 𝐶 < +∞ ) ) )
21 8 20 syl ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) → ( 𝐶 ∈ ℝ ↔ ( -∞ < 𝐶 ∧ 𝐶 < +∞ ) ) )
22 12 19 21 mpbir2and ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) → 𝐶 ∈ ℝ )
23 22 11 16 3jca ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) → ( 𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) )
24 23 ex ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) → ( 𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) )
25 rexr ⊢ ( 𝐶 ∈ ℝ → 𝐶 ∈ ℝ* )
26 25 3anim1i ⊢ ( ( 𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) → ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) )
27 24 26 impbid1 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ↔ ( 𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) )
28 4 27 bitrd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) )