Theorem disjin2 28782

Description: If a collection is disjoint, so is the collection of the intersections with a given set. (Contributed by Thierry Arnoux, 21-Jun-2020.)

Assertion

Ref	Expression
disjin2	⊢ (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐵 (𝐴 ∩ 𝐶))

Proof of Theorem disjin2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss2 3796	. . . . . . 7 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐶
2	1	sseli 3564	. . . . . 6 ⊢ (𝑦 ∈ (𝐴 ∩ 𝐶) → 𝑦 ∈ 𝐶)
3	2	anim2i 591	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ (𝐴 ∩ 𝐶)) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
4	3	ax-gen 1713	. . . 4 ⊢ ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ (𝐴 ∩ 𝐶)) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
5	4	rmoimi2 3376	. . 3 ⊢ (∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ∩ 𝐶))
6	5	alimi 1730	. 2 ⊢ (∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ∩ 𝐶))
7		df-disj 4554	. 2 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶)
8		df-disj 4554	. 2 ⊢ (Disj 𝑥 ∈ 𝐵 (𝐴 ∩ 𝐶) ↔ ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ∩ 𝐶))
9	6, 7, 8	3imtr4i 280	1 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐵 (𝐴 ∩ 𝐶))

Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473   ∈ wcel 1977  ∃*wrmo 2899   ∩ cin 3539  Disj wdisj 4553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rmo 2904  df-v 3175  df-in 3547  df-ss 3554  df-disj 4554

This theorem is referenced by:  ldgenpisyslem1  29553