Theorem disjeq2 5077

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 3972	. . . 4 ⊢ (𝐵 = 𝐶 → 𝐶 ⊆ 𝐵)
2	1	ralimi 3103	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)
3		disjss2 5076	. . 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 3971	. . . 4 ⊢ (𝐵 = 𝐶 → 𝐵 ⊆ 𝐶)
6	5	ralimi 3103	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
7		disjss2 5076	. . 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 3080   ⊆ wss 3900  Disj wdisj 5073

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2567  df-cleq 2755  df-clel 2839  df-ral 3081  df-rmo 3369  df-ss 3901  df-disj 5074

This theorem is used by:  disjeq2dv  5078  voliun  25902  carsgclctunlem2  35018  disjeq12i  37159  mblfinlem2  38767  voliunnfl  38773