Theorem disjin2 32456

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		elinel2 4194	. . . 4 ⊢ (𝑦 ∈ (𝐴 ∩ 𝐶) → 𝑦 ∈ 𝐶)
2	1	rmoimi 3734	. . 3 ⊢ (∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ∩ 𝐶))
3	2	alimi 1805	. 2 ⊢ (∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ∩ 𝐶))
4		df-disj 5115	. 2 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶)
5		df-disj 5115	. 2 ⊢ (Disj 𝑥 ∈ 𝐵 (𝐴 ∩ 𝐶) ↔ ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ∩ 𝐶))
6	3, 4, 5	3imtr4i 291	1 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐵 (𝐴 ∩ 𝐶))

Syntax hints:   → wi 4  ∀wal 1531   ∈ wcel 2098  ∃*wrmo 3362   ∩ cin 3943  Disj wdisj 5114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-mo 2528  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rmo 3363  df-v 3463  df-in 3951  df-disj 5115

This theorem is referenced by:  ldgenpisyslem1  33913