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Theorem disjss2 4655
Description: If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjss2 (∀𝑥𝐴 𝐵𝐶 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐴 𝐵))

Proof of Theorem disjss2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssel 3630 . . . . 5 (𝐵𝐶 → (𝑦𝐵𝑦𝐶))
21ralimi 2981 . . . 4 (∀𝑥𝐴 𝐵𝐶 → ∀𝑥𝐴 (𝑦𝐵𝑦𝐶))
3 rmoim 3440 . . . 4 (∀𝑥𝐴 (𝑦𝐵𝑦𝐶) → (∃*𝑥𝐴 𝑦𝐶 → ∃*𝑥𝐴 𝑦𝐵))
42, 3syl 17 . . 3 (∀𝑥𝐴 𝐵𝐶 → (∃*𝑥𝐴 𝑦𝐶 → ∃*𝑥𝐴 𝑦𝐵))
54alimdv 1885 . 2 (∀𝑥𝐴 𝐵𝐶 → (∀𝑦∃*𝑥𝐴 𝑦𝐶 → ∀𝑦∃*𝑥𝐴 𝑦𝐵))
6 df-disj 4653 . 2 (Disj 𝑥𝐴 𝐶 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐶)
7 df-disj 4653 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐵)
85, 6, 73imtr4g 285 1 (∀𝑥𝐴 𝐵𝐶 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐴 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1521  wcel 2030  wral 2941  ∃*wrmo 2944  wss 3607  Disj wdisj 4652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-ral 2946  df-rmo 2949  df-in 3614  df-ss 3621  df-disj 4653
This theorem is referenced by:  disjeq2  4656  0disj  4677  uniioombllem2  23397  uniioombllem4  23400  disjxwwlksn  26867  disjxwwlkn  26876  fusgreghash2wspv  27315  fsumiunss  40125
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