Metamath Proof Explorer


Theorem eqvrelcoss4

Description: Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 3-May-2019) (Revised by Peter Mazsa, 30-Sep-2021)

Ref Expression
Assertion eqvrelcoss4 ( EqvRel ≀ 𝑅 ↔ ∀ 𝑥𝑧 ( ( [ 𝑥 ] ≀ 𝑅 ∩ [ 𝑧 ] ≀ 𝑅 ) ≠ ∅ → ( [ 𝑥 ] 𝑅 ∩ [ 𝑧 ] 𝑅 ) ≠ ∅ ) )

Proof

Step Hyp Ref Expression
1 eqvrelcoss3 ( EqvRel ≀ 𝑅 ↔ ∀ 𝑥𝑦𝑧 ( ( 𝑥𝑅 𝑦𝑦𝑅 𝑧 ) → 𝑥𝑅 𝑧 ) )
2 trcoss2 ( ∀ 𝑥𝑦𝑧 ( ( 𝑥𝑅 𝑦𝑦𝑅 𝑧 ) → 𝑥𝑅 𝑧 ) ↔ ∀ 𝑥𝑧 ( ( [ 𝑥 ] ≀ 𝑅 ∩ [ 𝑧 ] ≀ 𝑅 ) ≠ ∅ → ( [ 𝑥 ] 𝑅 ∩ [ 𝑧 ] 𝑅 ) ≠ ∅ ) )
3 1 2 bitri ( EqvRel ≀ 𝑅 ↔ ∀ 𝑥𝑧 ( ( [ 𝑥 ] ≀ 𝑅 ∩ [ 𝑧 ] ≀ 𝑅 ) ≠ ∅ → ( [ 𝑥 ] 𝑅 ∩ [ 𝑧 ] 𝑅 ) ≠ ∅ ) )