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Theorem disjin 32389
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 4195 . . . 4 (𝑦 ∈ (𝐶𝐴) → 𝑦𝐶)
21rmoimi 3737 . . 3 (∃*𝑥𝐵 𝑦𝐶 → ∃*𝑥𝐵 𝑦 ∈ (𝐶𝐴))
32alimi 1806 . 2 (∀𝑦∃*𝑥𝐵 𝑦𝐶 → ∀𝑦∃*𝑥𝐵 𝑦 ∈ (𝐶𝐴))
4 df-disj 5114 . 2 (Disj 𝑥𝐵 𝐶 ↔ ∀𝑦∃*𝑥𝐵 𝑦𝐶)
5 df-disj 5114 . 2 (Disj 𝑥𝐵 (𝐶𝐴) ↔ ∀𝑦∃*𝑥𝐵 𝑦 ∈ (𝐶𝐴))
63, 4, 53imtr4i 292 1 (Disj 𝑥𝐵 𝐶Disj 𝑥𝐵 (𝐶𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1532  wcel 2099  ∃*wrmo 3372  cin 3946  Disj wdisj 5113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-mo 2530  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rmo 3373  df-v 3473  df-in 3954  df-disj 5114
This theorem is referenced by:  measinblem  33839  carsgclctunlem2  33939
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