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Theorem disjin2 30117
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 4055 . . . 4 (𝑦 ∈ (𝐴𝐶) → 𝑦𝐶)
21rmoimi 3639 . . 3 (∃*𝑥𝐵 𝑦𝐶 → ∃*𝑥𝐵 𝑦 ∈ (𝐴𝐶))
32alimi 1774 . 2 (∀𝑦∃*𝑥𝐵 𝑦𝐶 → ∀𝑦∃*𝑥𝐵 𝑦 ∈ (𝐴𝐶))
4 df-disj 4894 . 2 (Disj 𝑥𝐵 𝐶 ↔ ∀𝑦∃*𝑥𝐵 𝑦𝐶)
5 df-disj 4894 . 2 (Disj 𝑥𝐵 (𝐴𝐶) ↔ ∀𝑦∃*𝑥𝐵 𝑦 ∈ (𝐴𝐶))
63, 4, 53imtr4i 284 1 (Disj 𝑥𝐵 𝐶Disj 𝑥𝐵 (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1505  wcel 2050  ∃*wrmo 3085  cin 3822  Disj wdisj 4893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rmo 3090  df-v 3411  df-in 3830  df-disj 4894
This theorem is referenced by:  ldgenpisyslem1  31096
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