Description: Elementhood in the converse epsilon coset of A is elementhood in A . (Contributed by Peter Mazsa, 27-Jan-2019)
Ref | Expression | ||
---|---|---|---|
Assertion | eleccnvep | ⊢ ( 𝐴 ∈ 𝑉 → ( 𝐵 ∈ [ 𝐴 ] ◡ E ↔ 𝐵 ∈ 𝐴 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv | ⊢ Rel ◡ E | |
2 | relelec | ⊢ ( Rel ◡ E → ( 𝐵 ∈ [ 𝐴 ] ◡ E ↔ 𝐴 ◡ E 𝐵 ) ) | |
3 | 1 2 | ax-mp | ⊢ ( 𝐵 ∈ [ 𝐴 ] ◡ E ↔ 𝐴 ◡ E 𝐵 ) |
4 | brcnvep | ⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ◡ E 𝐵 ↔ 𝐵 ∈ 𝐴 ) ) | |
5 | 3 4 | syl5bb | ⊢ ( 𝐴 ∈ 𝑉 → ( 𝐵 ∈ [ 𝐴 ] ◡ E ↔ 𝐵 ∈ 𝐴 ) ) |