Theorem disjeq2 5078

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 3993	. . . 4 ⊢ (𝐵 = 𝐶 → 𝐶 ⊆ 𝐵)
2	1	ralimi 3101	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)
3		disjss2 5077	. . 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 3992	. . . 4 ⊢ (𝐵 = 𝐶 → 𝐵 ⊆ 𝐶)
6	5	ralimi 3101	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
7		disjss2 5077	. . 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 1570  ∀wral 3078   ⊆ wss 3902  Disj wdisj 5074

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 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2566  df-cleq 2754  df-clel 2837  df-ral 3079  df-rmo 3367  df-ss 3919  df-disj 5075

This theorem is used by:  disjeq2dv  5079  voliun  25783  carsgclctunlem2  34817  disjeq12i  36800  mblfinlem2  38394  voliunnfl  38400