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Theorem disjin 28781
 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 inss1 3795 . . . . . . 7 (𝐶𝐴) ⊆ 𝐶
21sseli 3564 . . . . . 6 (𝑦 ∈ (𝐶𝐴) → 𝑦𝐶)
32anim2i 591 . . . . 5 ((𝑥𝐵𝑦 ∈ (𝐶𝐴)) → (𝑥𝐵𝑦𝐶))
43ax-gen 1713 . . . 4 𝑥((𝑥𝐵𝑦 ∈ (𝐶𝐴)) → (𝑥𝐵𝑦𝐶))
54rmoimi2 3376 . . 3 (∃*𝑥𝐵 𝑦𝐶 → ∃*𝑥𝐵 𝑦 ∈ (𝐶𝐴))
65alimi 1730 . 2 (∀𝑦∃*𝑥𝐵 𝑦𝐶 → ∀𝑦∃*𝑥𝐵 𝑦 ∈ (𝐶𝐴))
7 df-disj 4554 . 2 (Disj 𝑥𝐵 𝐶 ↔ ∀𝑦∃*𝑥𝐵 𝑦𝐶)
8 df-disj 4554 . 2 (Disj 𝑥𝐵 (𝐶𝐴) ↔ ∀𝑦∃*𝑥𝐵 𝑦 ∈ (𝐶𝐴))
96, 7, 83imtr4i 280 1 (Disj 𝑥𝐵 𝐶Disj 𝑥𝐵 (𝐶𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473   ∈ wcel 1977  ∃*wrmo 2899   ∩ cin 3539  Disj wdisj 4553 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rmo 2904  df-v 3175  df-in 3547  df-ss 3554  df-disj 4554 This theorem is referenced by:  measinblem  29610  carsgclctunlem2  29708
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