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Theorem disjin2 30353
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 𝑥𝐵 𝐶Disj 𝑥𝐵 (𝐴𝐶))

Proof of Theorem disjin2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elinel2 4126 . . . 4 (𝑦 ∈ (𝐴𝐶) → 𝑦𝐶)
21rmoimi 3684 . . 3 (∃*𝑥𝐵 𝑦𝐶 → ∃*𝑥𝐵 𝑦 ∈ (𝐴𝐶))
32alimi 1813 . 2 (∀𝑦∃*𝑥𝐵 𝑦𝐶 → ∀𝑦∃*𝑥𝐵 𝑦 ∈ (𝐴𝐶))
4 df-disj 4999 . 2 (Disj 𝑥𝐵 𝐶 ↔ ∀𝑦∃*𝑥𝐵 𝑦𝐶)
5 df-disj 4999 . 2 (Disj 𝑥𝐵 (𝐴𝐶) ↔ ∀𝑦∃*𝑥𝐵 𝑦 ∈ (𝐴𝐶))
63, 4, 53imtr4i 295 1 (Disj 𝑥𝐵 𝐶Disj 𝑥𝐵 (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536  wcel 2112  ∃*wrmo 3112  cin 3883  Disj wdisj 4998
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 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-mo 2601  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-rmo 3117  df-v 3446  df-in 3891  df-disj 4999
This theorem is referenced by:  ldgenpisyslem1  31530
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