Theorem disjin2 33279

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 4102	. . . 4 ⊢ (𝑦 ∈ (𝐴 ∩ 𝐶) → 𝑦 ∈ 𝐶)
2	1	rmoimi 3703	. . 3 ⊢ (∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ∩ 𝐶))
3	2	alimi 1844	. 2 ⊢ (∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ∩ 𝐶))
4		df-disj 5076	. 2 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶)
5		df-disj 5076	. 2 ⊢ (Disj 𝑥 ∈ 𝐵 (𝐴 ∩ 𝐶) ↔ ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ∩ 𝐶))
6	3, 4, 5	3imtr4i 295	1 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐵 (𝐴 ∩ 𝐶))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568   ∈ wcel 2147  ∃*wrmo 3368   ∩ cin 4092  Disj wdisj 5075

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rmo 3369  df-v 3456  df-in 4094  df-disj 5076

This theorem is used by:  ldgenpisyslem1  34907