Theorem disjabso

Description: Disjointness is absolute for transitive models. Compare Example I.16.3 of Kunen2 p. 96 and the following discussion. (Contributed by Eric Schmidt, 19-Oct-2025)

	Ref	Expression
Assertion	disjabso	⊢ ( ( Tr 𝑀 ∧ 𝐴 ∈ 𝑀 ) → ( ( 𝐴 ∩ 𝐵 ) = ∅ ↔ ∀ 𝑥 ∈ 𝑀 ( 𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		disj	⊢ ( ( 𝐴 ∩ 𝐵 ) = ∅ ↔ ∀ 𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 )
2		ralabso	⊢ ( ( Tr 𝑀 ∧ 𝐴 ∈ 𝑀 ) → ( ∀ 𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ↔ ∀ 𝑥 ∈ 𝑀 ( 𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵 ) ) )
3	1 2	bitrid	⊢ ( ( Tr 𝑀 ∧ 𝐴 ∈ 𝑀 ) → ( ( 𝐴 ∩ 𝐵 ) = ∅ ↔ ∀ 𝑥 ∈ 𝑀 ( 𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵 ) ) )