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Theorem disjss2 5047
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 3919 . . . . 5 (𝐵𝐶 → (𝑦𝐵𝑦𝐶))
21ralimi 3089 . . . 4 (∀𝑥𝐴 𝐵𝐶 → ∀𝑥𝐴 (𝑦𝐵𝑦𝐶))
3 rmoim 3679 . . . 4 (∀𝑥𝐴 (𝑦𝐵𝑦𝐶) → (∃*𝑥𝐴 𝑦𝐶 → ∃*𝑥𝐴 𝑦𝐵))
42, 3syl 17 . . 3 (∀𝑥𝐴 𝐵𝐶 → (∃*𝑥𝐴 𝑦𝐶 → ∃*𝑥𝐴 𝑦𝐵))
54alimdv 1923 . 2 (∀𝑥𝐴 𝐵𝐶 → (∀𝑦∃*𝑥𝐴 𝑦𝐶 → ∀𝑦∃*𝑥𝐴 𝑦𝐵))
6 df-disj 5045 . 2 (Disj 𝑥𝐴 𝐶 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐶)
7 df-disj 5045 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐵)
85, 6, 73imtr4g 296 1 (∀𝑥𝐴 𝐵𝐶 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐴 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540  wcel 2110  wral 3066  ∃*wrmo 3069  wss 3892  Disj wdisj 5044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1545  df-ex 1787  df-sb 2072  df-mo 2542  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rmo 3074  df-v 3433  df-in 3899  df-ss 3909  df-disj 5045
This theorem is referenced by:  disjeq2  5048  0disj  5071  uniioombllem2  24745  uniioombllem4  24748  disjxwwlksn  28265  disjxwwlkn  28274  fusgreghash2wspv  28695  fsumiunss  43087
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