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Theorem disjin2 29268
 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 3783 . . . . . 6 (𝑦 ∈ (𝐴𝐶) → 𝑦𝐶)
21anim2i 592 . . . . 5 ((𝑥𝐵𝑦 ∈ (𝐴𝐶)) → (𝑥𝐵𝑦𝐶))
32ax-gen 1719 . . . 4 𝑥((𝑥𝐵𝑦 ∈ (𝐴𝐶)) → (𝑥𝐵𝑦𝐶))
43rmoimi2 3395 . . 3 (∃*𝑥𝐵 𝑦𝐶 → ∃*𝑥𝐵 𝑦 ∈ (𝐴𝐶))
54alimi 1736 . 2 (∀𝑦∃*𝑥𝐵 𝑦𝐶 → ∀𝑦∃*𝑥𝐵 𝑦 ∈ (𝐴𝐶))
6 df-disj 4589 . 2 (Disj 𝑥𝐵 𝐶 ↔ ∀𝑦∃*𝑥𝐵 𝑦𝐶)
7 df-disj 4589 . 2 (Disj 𝑥𝐵 (𝐴𝐶) ↔ ∀𝑦∃*𝑥𝐵 𝑦 ∈ (𝐴𝐶))
85, 6, 73imtr4i 281 1 (Disj 𝑥𝐵 𝐶Disj 𝑥𝐵 (𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1478   ∈ wcel 1987  ∃*wrmo 2910   ∩ cin 3558  Disj wdisj 4588 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rmo 2915  df-v 3191  df-in 3566  df-disj 4589 This theorem is referenced by:  ldgenpisyslem1  30031
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