Theorem disjin 32841

Description: If a collection is disjoint, so is the collection of the intersections with a given set. (Contributed by Thierry Arnoux, 14-Feb-2017.)

Assertion

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

Proof of Theorem disjin

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elinel1 4156	. . . 4 ⊢ (𝑦 ∈ (𝐶 ∩ 𝐴) → 𝑦 ∈ 𝐶)
2	1	rmoimi 3708	. . 3 ⊢ (∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐶 ∩ 𝐴))
3	2	alimi 1834	. 2 ⊢ (∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐶 ∩ 𝐴))
4		df-disj 5073	. 2 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶)
5		df-disj 5073	. 2 ⊢ (Disj 𝑥 ∈ 𝐵 (𝐶 ∩ 𝐴) ↔ ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐶 ∩ 𝐴))
6	3, 4, 5	3imtr4i 295	1 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐵 (𝐶 ∩ 𝐴))

Syntax hints:   → wi 4  ∀wal 1561   ∈ wcel 2145  ∃*wrmo 3369   ∩ cin 3906  Disj wdisj 5072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-mo 2569  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rmo 3370  df-v 3459  df-in 3914  df-disj 5073

This theorem is referenced by:  measinblem  34527  carsgclctunlem2  34626