Theorem eliniseg

Description: Membership in an initial segment. The idiom (`' A " { B } ) , meaning { x | x A B } ` , is used to specify an initial segment in (for example) Definition 6.21 of TakeutiZaring p. 30. (Contributed by NM, 28-Apr-2004) (Proof shortened by Andrew Salmon, 27-Aug-2011)

	Ref	Expression
Hypothesis	eliniseg.1	⊢ 𝐶 ∈ V
Assertion	eliniseg	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐶 ∈ ( ◡ 𝐴 “ { 𝐵 } ) ↔ 𝐶 𝐴 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		eliniseg.1	⊢ 𝐶 ∈ V
2		elimasng	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V ) → ( 𝐶 ∈ ( ◡ 𝐴 “ { 𝐵 } ) ↔ ⟨ 𝐵 , 𝐶 ⟩ ∈ ◡ 𝐴 ) )
3		df-br	⊢ ( 𝐵 ◡ 𝐴 𝐶 ↔ ⟨ 𝐵 , 𝐶 ⟩ ∈ ◡ 𝐴 )
4	2 3	syl6bbr	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V ) → ( 𝐶 ∈ ( ◡ 𝐴 “ { 𝐵 } ) ↔ 𝐵 ◡ 𝐴 𝐶 ) )
5		brcnvg	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V ) → ( 𝐵 ◡ 𝐴 𝐶 ↔ 𝐶 𝐴 𝐵 ) )
6	4 5	bitrd	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V ) → ( 𝐶 ∈ ( ◡ 𝐴 “ { 𝐵 } ) ↔ 𝐶 𝐴 𝐵 ) )
7	1 6	mpan2	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐶 ∈ ( ◡ 𝐴 “ { 𝐵 } ) ↔ 𝐶 𝐴 𝐵 ) )