Theorem inecmo2

Description: Equivalence of a double restricted universal quantification and a restricted "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 29-May-2018) (Revised by Peter Mazsa, 2-Sep-2021)

	Ref	Expression
Assertion	inecmo2	⊢ ( ( ∀ 𝑢 ∈ 𝐴 ∀ 𝑣 ∈ 𝐴 ( 𝑢 = 𝑣 ∨ ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) = ∅ ) ∧ Rel 𝑅 ) ↔ ( ∀ 𝑥 ∃* 𝑢 ∈ 𝐴 𝑢 𝑅 𝑥 ∧ Rel 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( 𝑢 = 𝑣 → 𝑢 = 𝑣 )
2	1	inecmo	⊢ ( Rel 𝑅 → ( ∀ 𝑢 ∈ 𝐴 ∀ 𝑣 ∈ 𝐴 ( 𝑢 = 𝑣 ∨ ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) = ∅ ) ↔ ∀ 𝑥 ∃* 𝑢 ∈ 𝐴 𝑢 𝑅 𝑥 ) )
3	2	pm5.32ri	⊢ ( ( ∀ 𝑢 ∈ 𝐴 ∀ 𝑣 ∈ 𝐴 ( 𝑢 = 𝑣 ∨ ( [ 𝑢 ] 𝑅 ∩ [ 𝑣 ] 𝑅 ) = ∅ ) ∧ Rel 𝑅 ) ↔ ( ∀ 𝑥 ∃* 𝑢 ∈ 𝐴 𝑢 𝑅 𝑥 ∧ Rel 𝑅 ) )