Theorem disjss2 3739

Description: If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
disjss2	⊢ (∀x ∈ A B ⊆ 𝐶 → (Disj x ∈ A 𝐶 → Disj x ∈ A B))

Proof of Theorem disjss2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 2933	. . . . 5 ⊢ (B ⊆ 𝐶 → (y ∈ B → y ∈ 𝐶))
2	1	ralimi 2378	. . . 4 ⊢ (∀x ∈ A B ⊆ 𝐶 → ∀x ∈ A (y ∈ B → y ∈ 𝐶))
3		rmoim 2734	. . . 4 ⊢ (∀x ∈ A (y ∈ B → y ∈ 𝐶) → (∃*x ∈ A y ∈ 𝐶 → ∃*x ∈ A y ∈ B))
4	2, 3	syl 14	. . 3 ⊢ (∀x ∈ A B ⊆ 𝐶 → (∃*x ∈ A y ∈ 𝐶 → ∃*x ∈ A y ∈ B))
5	4	alimdv 1756	. 2 ⊢ (∀x ∈ A B ⊆ 𝐶 → (∀y∃*x ∈ A y ∈ 𝐶 → ∀y∃*x ∈ A y ∈ B))
6		df-disj 3737	. 2 ⊢ (Disj x ∈ A 𝐶 ↔ ∀y∃*x ∈ A y ∈ 𝐶)
7		df-disj 3737	. 2 ⊢ (Disj x ∈ A B ↔ ∀y∃*x ∈ A y ∈ B)
8	5, 6, 7	3imtr4g 194	1 ⊢ (∀x ∈ A B ⊆ 𝐶 → (Disj x ∈ A 𝐶 → Disj x ∈ A B))

Syntax hints:   → wi 4  ∀wal 1240   ∈ wcel 1390  ∀wral 2300  ∃*wrmo 2303   ⊆ 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-bndl 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:  disjeq2  3740  0disj  3752