Theorem disjss1 3742

Description: A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
disjss1	⊢ (A ⊆ B → (Disj x ∈ B 𝐶 → Disj x ∈ A 𝐶))

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   𝐶(x)

Proof of Theorem disjss1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 2933	. . . . . 6 ⊢ (A ⊆ B → (x ∈ A → x ∈ B))
2	1	anim1d 319	. . . . 5 ⊢ (A ⊆ B → ((x ∈ A ∧ y ∈ 𝐶) → (x ∈ B ∧ y ∈ 𝐶)))
3	2	alrimiv 1751	. . . 4 ⊢ (A ⊆ B → ∀x((x ∈ A ∧ y ∈ 𝐶) → (x ∈ B ∧ y ∈ 𝐶)))
4		moim 1961	. . . 4 ⊢ (∀x((x ∈ A ∧ y ∈ 𝐶) → (x ∈ B ∧ y ∈ 𝐶)) → (∃*x(x ∈ B ∧ y ∈ 𝐶) → ∃*x(x ∈ A ∧ y ∈ 𝐶)))
5	3, 4	syl 14	. . 3 ⊢ (A ⊆ B → (∃*x(x ∈ B ∧ y ∈ 𝐶) → ∃*x(x ∈ A ∧ y ∈ 𝐶)))
6	5	alimdv 1756	. 2 ⊢ (A ⊆ B → (∀y∃*x(x ∈ B ∧ y ∈ 𝐶) → ∀y∃*x(x ∈ A ∧ y ∈ 𝐶)))
7		dfdisj2 3738	. 2 ⊢ (Disj x ∈ B 𝐶 ↔ ∀y∃*x(x ∈ B ∧ y ∈ 𝐶))
8		dfdisj2 3738	. 2 ⊢ (Disj x ∈ A 𝐶 ↔ ∀y∃*x(x ∈ A ∧ y ∈ 𝐶))
9	6, 7, 8	3imtr4g 194	1 ⊢ (A ⊆ B → (Disj x ∈ B 𝐶 → Disj x ∈ A 𝐶))

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1240   ∈ wcel 1390  ∃*wmo 1898   ⊆ 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-rmo 2308  df-in 2918  df-ss 2925  df-disj 3737

This theorem is referenced by:  disjeq1  3743  disjx0  3754