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Theorem disjin 30344
 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 4146 . . . 4 (𝑦 ∈ (𝐶𝐴) → 𝑦𝐶)
21rmoimi 3708 . . 3 (∃*𝑥𝐵 𝑦𝐶 → ∃*𝑥𝐵 𝑦 ∈ (𝐶𝐴))
32alimi 1813 . 2 (∀𝑦∃*𝑥𝐵 𝑦𝐶 → ∀𝑦∃*𝑥𝐵 𝑦 ∈ (𝐶𝐴))
4 df-disj 5008 . 2 (Disj 𝑥𝐵 𝐶 ↔ ∀𝑦∃*𝑥𝐵 𝑦𝐶)
5 df-disj 5008 . 2 (Disj 𝑥𝐵 (𝐶𝐴) ↔ ∀𝑦∃*𝑥𝐵 𝑦 ∈ (𝐶𝐴))
63, 4, 53imtr4i 295 1 (Disj 𝑥𝐵 𝐶Disj 𝑥𝐵 (𝐶𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536   ∈ wcel 2114  ∃*wrmo 3133   ∩ cin 3907  Disj wdisj 5007 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-mo 2622  df-clab 2801  df-cleq 2815  df-clel 2894  df-ral 3135  df-rmo 3138  df-v 3471  df-in 3915  df-disj 5008 This theorem is referenced by:  measinblem  31553  carsgclctunlem2  31651
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