Theorem disjabso 45489

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.)

Assertion

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

Distinct variable groups:   𝑥,𝑀   𝑥,𝐴   𝑥,𝐵

Proof of Theorem disjabso

Step	Hyp	Ref	Expression
1		disj 4394	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
2		ralabso 45482	. 2 ⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)))
3	1, 2	bitrid 285	1 ⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1550   ∈ wcel 2132  ∀wral 3066   ∩ cin 3894  ∅c0 4276  Tr wtr 5197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-ext 2724

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1553  df-fal 1563  df-ex 1790  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-ral 3067  df-v 3446  df-dif 3898  df-in 3902  df-ss 3912  df-nul 4277  df-uni 4856  df-tr 5198

This theorem is referenced by:  modelac8prim  45506