Theorem eleccossin

Description: Two ways of saying that the coset of A and the coset of C have the common element B . (Contributed by Peter Mazsa, 15-Oct-2021)

	Ref	Expression
Assertion	eleccossin	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( 𝐵 ∈ ( [ 𝐴 ] ≀ 𝑅 ∩ [ 𝐶 ] ≀ 𝑅 ) ↔ ( 𝐴 ≀ 𝑅 𝐵 ∧ 𝐵 ≀ 𝑅 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		elin	⊢ ( 𝐵 ∈ ( [ 𝐴 ] ≀ 𝑅 ∩ [ 𝐶 ] ≀ 𝑅 ) ↔ ( 𝐵 ∈ [ 𝐴 ] ≀ 𝑅 ∧ 𝐵 ∈ [ 𝐶 ] ≀ 𝑅 ) )
2		relcoss	⊢ Rel ≀ 𝑅
3		relelec	⊢ ( Rel ≀ 𝑅 → ( 𝐵 ∈ [ 𝐴 ] ≀ 𝑅 ↔ 𝐴 ≀ 𝑅 𝐵 ) )
4	2 3	ax-mp	⊢ ( 𝐵 ∈ [ 𝐴 ] ≀ 𝑅 ↔ 𝐴 ≀ 𝑅 𝐵 )
5		relelec	⊢ ( Rel ≀ 𝑅 → ( 𝐵 ∈ [ 𝐶 ] ≀ 𝑅 ↔ 𝐶 ≀ 𝑅 𝐵 ) )
6	2 5	ax-mp	⊢ ( 𝐵 ∈ [ 𝐶 ] ≀ 𝑅 ↔ 𝐶 ≀ 𝑅 𝐵 )
7	4 6	anbi12i	⊢ ( ( 𝐵 ∈ [ 𝐴 ] ≀ 𝑅 ∧ 𝐵 ∈ [ 𝐶 ] ≀ 𝑅 ) ↔ ( 𝐴 ≀ 𝑅 𝐵 ∧ 𝐶 ≀ 𝑅 𝐵 ) )
8	1 7	bitri	⊢ ( 𝐵 ∈ ( [ 𝐴 ] ≀ 𝑅 ∩ [ 𝐶 ] ≀ 𝑅 ) ↔ ( 𝐴 ≀ 𝑅 𝐵 ∧ 𝐶 ≀ 𝑅 𝐵 ) )
9		brcosscnvcoss	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( 𝐵 ≀ 𝑅 𝐶 ↔ 𝐶 ≀ 𝑅 𝐵 ) )
10	9	anbi2d	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( ( 𝐴 ≀ 𝑅 𝐵 ∧ 𝐵 ≀ 𝑅 𝐶 ) ↔ ( 𝐴 ≀ 𝑅 𝐵 ∧ 𝐶 ≀ 𝑅 𝐵 ) ) )
11	8 10	bitr4id	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( 𝐵 ∈ ( [ 𝐴 ] ≀ 𝑅 ∩ [ 𝐶 ] ≀ 𝑅 ) ↔ ( 𝐴 ≀ 𝑅 𝐵 ∧ 𝐵 ≀ 𝑅 𝐶 ) ) )