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Metamath Proof Explorer

Theorem brcoss2

Description: Alternate form of the A and B are cosets by R binary relation. (Contributed by Peter Mazsa, 26-Mar-2019)

Ref Expression
Assertion brcoss2 ⊢ A ∈ V ∧ B ∈ W → A ≀ R B ↔ ∃ u A ∈ u R ∧ B ∈ u R

Proof

Step Hyp Ref Expression
1 brcoss ⊢ A ∈ V ∧ B ∈ W → A ≀ R B ↔ ∃ u u R A ∧ u R B
2 exan3 ⊢ A ∈ V ∧ B ∈ W → ∃ u A ∈ u R ∧ B ∈ u R ↔ ∃ u u R A ∧ u R B
3 1 2 bitr4d ⊢ A ∈ V ∧ B ∈ W → A ≀ R B ↔ ∃ u A ∈ u R ∧ B ∈ u R