Theorem disjeq2 5079

Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
disjeq2	⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑥 ∈ 𝐴 𝐶))

Proof of Theorem disjeq2

Step	Hyp	Ref	Expression
1		eqimss2 3995	. . . 4 ⊢ (𝐵 = 𝐶 → 𝐶 ⊆ 𝐵)
2	1	ralimi 3101	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)
3		disjss2 5078	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑥 ∈ 𝐴 𝐶))
4	2, 3	syl 18	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑥 ∈ 𝐴 𝐶))
5		eqimss 3994	. . . 4 ⊢ (𝐵 = 𝐶 → 𝐵 ⊆ 𝐶)
6	5	ralimi 3101	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
7		disjss2 5078	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (Disj 𝑥 ∈ 𝐴 𝐶 → Disj 𝑥 ∈ 𝐴 𝐵))
8	6, 7	syl 18	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → (Disj 𝑥 ∈ 𝐴 𝐶 → Disj 𝑥 ∈ 𝐴 𝐵))
9	4, 8	impbid 215	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑥 ∈ 𝐴 𝐶))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569  ∀wral 3078   ⊆ wss 3904  Disj wdisj 5075

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-mo 2566  df-cleq 2754  df-clel 2837  df-ral 3079  df-rmo 3368  df-ss 3921  df-disj 5076

This theorem is used by:  disjeq2dv  5080  voliun  25724  carsgclctunlem2  34718  disjeq12i  36733  mblfinlem2  38337  voliunnfl  38343