Metamath Proof Explorer

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 -1 B ↔ C A B

Proof

Step Hyp Ref Expression
1 eliniseg.1 ⊢ C ∈ V
2 elimasng ⊢ B ∈ V ∧ C ∈ V → C ∈ A -1 B ↔ B C ∈ A -1
3 df-br ⊢ B A -1 C ↔ B C ∈ A -1
4 2 3 syl6bbr ⊢ B ∈ V ∧ C ∈ V → C ∈ A -1 B ↔ B A -1 C
5 brcnvg ⊢ B ∈ V ∧ C ∈ V → B A -1 C ↔ C A B
6 4 5 bitrd ⊢ B ∈ V ∧ C ∈ V → C ∈ A -1 B ↔ C A B
7 1 6 mpan2 ⊢ B ∈ V → C ∈ A -1 B ↔ C A B