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Metamath Proof Explorer

Theorem snunico

Description: The closure of the open end of a right-open real interval. (Contributed by Mario Carneiro, 16-Jun-2014)

		Ref	Expression
	Assertion	snunico	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ) → ( ( 𝐴 [,) 𝐵 ) ∪ { 𝐵 } ) = ( 𝐴 [,] 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		simp2	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ) → 𝐵 ∈ ℝ^* )
2		iccid	⊢ ( 𝐵 ∈ ℝ^* → ( 𝐵 [,] 𝐵 ) = { 𝐵 } )
3	1 2	syl	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ) → ( 𝐵 [,] 𝐵 ) = { 𝐵 } )
4	3	uneq2d	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ) → ( ( 𝐴 [,) 𝐵 ) ∪ ( 𝐵 [,] 𝐵 ) ) = ( ( 𝐴 [,) 𝐵 ) ∪ { 𝐵 } ) )
5		simp1	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ∈ ℝ^* )
6		simp3	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ≤ 𝐵 )
7	1	xrleidd	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ) → 𝐵 ≤ 𝐵 )
8		df-ico	⊢ [,) = ( 𝑥 ∈ ℝ^* , 𝑦 ∈ ℝ^* ↦ { 𝑧 ∈ ℝ^* ∣ ( 𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦 ) } )
9		df-icc	⊢ [,] = ( 𝑥 ∈ ℝ^* , 𝑦 ∈ ℝ^* ↦ { 𝑧 ∈ ℝ^* ∣ ( 𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦 ) } )
10		xrlenlt	⊢ ( ( 𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^* ) → ( 𝐵 ≤ 𝑤 ↔ ¬ 𝑤 < 𝐵 ) )
11		xrltle	⊢ ( ( 𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ) → ( 𝑤 < 𝐵 → 𝑤 ≤ 𝐵 ) )
12	11	3adant3	⊢ ( ( 𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ) → ( 𝑤 < 𝐵 → 𝑤 ≤ 𝐵 ) )
13	12	adantrd	⊢ ( ( 𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ) → ( ( 𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐵 ) → 𝑤 ≤ 𝐵 ) )
14		xrletr	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^* ) → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝑤 ) → 𝐴 ≤ 𝑤 ) )
15	8 9 10 9 13 14	ixxun	⊢ ( ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ) ∧ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵 ) ) → ( ( 𝐴 [,) 𝐵 ) ∪ ( 𝐵 [,] 𝐵 ) ) = ( 𝐴 [,] 𝐵 ) )
16	5 1 1 6 7 15	syl32anc	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ) → ( ( 𝐴 [,) 𝐵 ) ∪ ( 𝐵 [,] 𝐵 ) ) = ( 𝐴 [,] 𝐵 ) )
17	4 16	eqtr3d	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ) → ( ( 𝐴 [,) 𝐵 ) ∪ { 𝐵 } ) = ( 𝐴 [,] 𝐵 ) )