Theorem disjin 30925

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 4129	. . . 4 ⊢ (𝑦 ∈ (𝐶 ∩ 𝐴) → 𝑦 ∈ 𝐶)
2	1	rmoimi 3677	. . 3 ⊢ (∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐶 ∩ 𝐴))
3	2	alimi 1814	. 2 ⊢ (∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐶 ∩ 𝐴))
4		df-disj 5040	. 2 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶)
5		df-disj 5040	. 2 ⊢ (Disj 𝑥 ∈ 𝐵 (𝐶 ∩ 𝐴) ↔ ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐶 ∩ 𝐴))
6	3, 4, 5	3imtr4i 292	1 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐵 (𝐶 ∩ 𝐴))

Syntax hints:   → wi 4  ∀wal 1537   ∈ wcel 2106  ∃*wrmo 3067   ∩ cin 3886  Disj wdisj 5039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rmo 3071  df-v 3434  df-in 3894  df-disj 5040

This theorem is referenced by:  measinblem  32188  carsgclctunlem2  32286