Theorem disjeq1 3743

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

Assertion

Ref	Expression
disjeq1	⊢ (A = B → (Disj x ∈ A 𝐶 ↔ Disj x ∈ B 𝐶))

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   𝐶(x)

Proof of Theorem disjeq1

Step	Hyp	Ref	Expression
1		eqimss2 2992	. . 3 ⊢ (A = B → B ⊆ A)
2		disjss1 3742	. . 3 ⊢ (B ⊆ A → (Disj x ∈ A 𝐶 → Disj x ∈ B 𝐶))
3	1, 2	syl 14	. 2 ⊢ (A = B → (Disj x ∈ A 𝐶 → Disj x ∈ B 𝐶))
4		eqimss 2991	. . 3 ⊢ (A = B → A ⊆ B)
5		disjss1 3742	. . 3 ⊢ (A ⊆ B → (Disj x ∈ B 𝐶 → Disj x ∈ A 𝐶))
6	4, 5	syl 14	. 2 ⊢ (A = B → (Disj x ∈ B 𝐶 → Disj x ∈ A 𝐶))
7	3, 6	impbid 120	1 ⊢ (A = B → (Disj x ∈ A 𝐶 ↔ Disj x ∈ B 𝐶))

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242   ⊆ 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-rmo 2308  df-in 2918  df-ss 2925  df-disj 3737

This theorem is referenced by:  disjeq1d  3744