Theorem brcoss3

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

	Ref	Expression
Assertion	brcoss3	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ≀ 𝑅 𝐵 ↔ ( [ 𝐴 ] ◡ 𝑅 ∩ [ 𝐵 ] ◡ 𝑅 ) ≠ ∅ ) )

Proof

Step	Hyp	Ref	Expression
1		brcnvg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑢 ∈ V ) → ( 𝐴 ◡ 𝑅 𝑢 ↔ 𝑢 𝑅 𝐴 ) )
2	1	elvd	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ◡ 𝑅 𝑢 ↔ 𝑢 𝑅 𝐴 ) )
3		brcnvg	⊢ ( ( 𝐵 ∈ 𝑊 ∧ 𝑢 ∈ V ) → ( 𝐵 ◡ 𝑅 𝑢 ↔ 𝑢 𝑅 𝐵 ) )
4	3	elvd	⊢ ( 𝐵 ∈ 𝑊 → ( 𝐵 ◡ 𝑅 𝑢 ↔ 𝑢 𝑅 𝐵 ) )
5	2 4	bi2anan9	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ( 𝐴 ◡ 𝑅 𝑢 ∧ 𝐵 ◡ 𝑅 𝑢 ) ↔ ( 𝑢 𝑅 𝐴 ∧ 𝑢 𝑅 𝐵 ) ) )
6	5	exbidv	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∃ 𝑢 ( 𝐴 ◡ 𝑅 𝑢 ∧ 𝐵 ◡ 𝑅 𝑢 ) ↔ ∃ 𝑢 ( 𝑢 𝑅 𝐴 ∧ 𝑢 𝑅 𝐵 ) ) )
7		ecinn0	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ( [ 𝐴 ] ◡ 𝑅 ∩ [ 𝐵 ] ◡ 𝑅 ) ≠ ∅ ↔ ∃ 𝑢 ( 𝐴 ◡ 𝑅 𝑢 ∧ 𝐵 ◡ 𝑅 𝑢 ) ) )
8		brcoss	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ≀ 𝑅 𝐵 ↔ ∃ 𝑢 ( 𝑢 𝑅 𝐴 ∧ 𝑢 𝑅 𝐵 ) ) )
9	6 7 8	3bitr4rd	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ≀ 𝑅 𝐵 ↔ ( [ 𝐴 ] ◡ 𝑅 ∩ [ 𝐵 ] ◡ 𝑅 ) ≠ ∅ ) )