Theorem fzosplitsni

Description: Membership in a half-open range extended by a singleton. (Contributed by Stefan O'Rear, 23-Aug-2015)

	Ref	Expression
Assertion	fzosplitsni	⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) → ( 𝐶 ∈ ( 𝐴 ..^ ( 𝐵 + 1 ) ) ↔ ( 𝐶 ∈ ( 𝐴 ..^ 𝐵 ) ∨ 𝐶 = 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		fzosplitsn	⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) → ( 𝐴 ..^ ( 𝐵 + 1 ) ) = ( ( 𝐴 ..^ 𝐵 ) ∪ { 𝐵 } ) )
2	1	eleq2d	⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) → ( 𝐶 ∈ ( 𝐴 ..^ ( 𝐵 + 1 ) ) ↔ 𝐶 ∈ ( ( 𝐴 ..^ 𝐵 ) ∪ { 𝐵 } ) ) )
3		elun	⊢ ( 𝐶 ∈ ( ( 𝐴 ..^ 𝐵 ) ∪ { 𝐵 } ) ↔ ( 𝐶 ∈ ( 𝐴 ..^ 𝐵 ) ∨ 𝐶 ∈ { 𝐵 } ) )
4		elsn2g	⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) → ( 𝐶 ∈ { 𝐵 } ↔ 𝐶 = 𝐵 ) )
5	4	orbi2d	⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) → ( ( 𝐶 ∈ ( 𝐴 ..^ 𝐵 ) ∨ 𝐶 ∈ { 𝐵 } ) ↔ ( 𝐶 ∈ ( 𝐴 ..^ 𝐵 ) ∨ 𝐶 = 𝐵 ) ) )
6	3 5	syl5bb	⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) → ( 𝐶 ∈ ( ( 𝐴 ..^ 𝐵 ) ∪ { 𝐵 } ) ↔ ( 𝐶 ∈ ( 𝐴 ..^ 𝐵 ) ∨ 𝐶 = 𝐵 ) ) )
7	2 6	bitrd	⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) → ( 𝐶 ∈ ( 𝐴 ..^ ( 𝐵 + 1 ) ) ↔ ( 𝐶 ∈ ( 𝐴 ..^ 𝐵 ) ∨ 𝐶 = 𝐵 ) ) )