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Theorem disjin 29962
 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 𝑥𝐵 𝐶Disj 𝑥𝐵 (𝐶𝐴))

Proof of Theorem disjin
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elinel1 4021 . . . . . 6 (𝑦 ∈ (𝐶𝐴) → 𝑦𝐶)
21anim2i 610 . . . . 5 ((𝑥𝐵𝑦 ∈ (𝐶𝐴)) → (𝑥𝐵𝑦𝐶))
32ax-gen 1839 . . . 4 𝑥((𝑥𝐵𝑦 ∈ (𝐶𝐴)) → (𝑥𝐵𝑦𝐶))
43rmoimi2 3622 . . 3 (∃*𝑥𝐵 𝑦𝐶 → ∃*𝑥𝐵 𝑦 ∈ (𝐶𝐴))
54alimi 1855 . 2 (∀𝑦∃*𝑥𝐵 𝑦𝐶 → ∀𝑦∃*𝑥𝐵 𝑦 ∈ (𝐶𝐴))
6 df-disj 4855 . 2 (Disj 𝑥𝐵 𝐶 ↔ ∀𝑦∃*𝑥𝐵 𝑦𝐶)
7 df-disj 4855 . 2 (Disj 𝑥𝐵 (𝐶𝐴) ↔ ∀𝑦∃*𝑥𝐵 𝑦 ∈ (𝐶𝐴))
85, 6, 73imtr4i 284 1 (Disj 𝑥𝐵 𝐶Disj 𝑥𝐵 (𝐶𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386  ∀wal 1599   ∈ wcel 2106  ∃*wrmo 3092   ∩ cin 3790  Disj wdisj 4854 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-rmo 3097  df-v 3399  df-in 3798  df-disj 4855 This theorem is referenced by:  measinblem  30881  carsgclctunlem2  30979
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