Theorem disjabso 45902

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 4402	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
2		ralabso 45895	. 2 ⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)))
3	1, 2	bitrid 286	1 ⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ∩ cin 3897  ∅c0 4278  Tr wtr 5211

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-v 3452  df-dif 3901  df-in 3905  df-ss 3915  df-nul 4279  df-uni 4867  df-tr 5212

This theorem is used by:  modelac8prim  45919