Theorem disjeq2 3740

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 2992	. . . 4 ⊢ (B = 𝐶 → 𝐶 ⊆ B)
2	1	ralimi 2378	. . 3 ⊢ (∀x ∈ A B = 𝐶 → ∀x ∈ A 𝐶 ⊆ B)
3		disjss2 3739	. . 3 ⊢ (∀x ∈ A 𝐶 ⊆ B → (Disj x ∈ A B → Disj x ∈ A 𝐶))
4	2, 3	syl 14	. 2 ⊢ (∀x ∈ A B = 𝐶 → (Disj x ∈ A B → Disj x ∈ A 𝐶))
5		eqimss 2991	. . . 4 ⊢ (B = 𝐶 → B ⊆ 𝐶)
6	5	ralimi 2378	. . 3 ⊢ (∀x ∈ A B = 𝐶 → ∀x ∈ A B ⊆ 𝐶)
7		disjss2 3739	. . 3 ⊢ (∀x ∈ A B ⊆ 𝐶 → (Disj x ∈ A 𝐶 → Disj x ∈ A B))
8	6, 7	syl 14	. 2 ⊢ (∀x ∈ A B = 𝐶 → (Disj x ∈ A 𝐶 → Disj x ∈ A B))
9	4, 8	impbid 120	1 ⊢ (∀x ∈ A B = 𝐶 → (Disj x ∈ A B ↔ Disj x ∈ A 𝐶))

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242  ∀wral 2300   ⊆ wss 2911  Disj wdisj 3736

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-ral 2305  df-rmo 2308  df-in 2918  df-ss 2925  df-disj 3737

This theorem is referenced by:  disjeq2dv  3741