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Theorem disjss1f 28768
 Description: A subset of a disjoint collection is disjoint. (Contributed by Thierry Arnoux, 6-Apr-2017.)
Hypotheses
Ref Expression
disjss1f.1 𝑥𝐴
disjss1f.2 𝑥𝐵
Assertion
Ref Expression
disjss1f (𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))

Proof of Theorem disjss1f
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 disjss1f.1 . . . 4 𝑥𝐴
2 disjss1f.2 . . . 4 𝑥𝐵
31, 2ssrmo 28718 . . 3 (𝐴𝐵 → (∃*𝑥𝐵 𝑦𝐶 → ∃*𝑥𝐴 𝑦𝐶))
43alimdv 1832 . 2 (𝐴𝐵 → (∀𝑦∃*𝑥𝐵 𝑦𝐶 → ∀𝑦∃*𝑥𝐴 𝑦𝐶))
5 df-disj 4554 . 2 (Disj 𝑥𝐵 𝐶 ↔ ∀𝑦∃*𝑥𝐵 𝑦𝐶)
6 df-disj 4554 . 2 (Disj 𝑥𝐴 𝐶 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐶)
74, 5, 63imtr4g 284 1 (𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473   ∈ wcel 1977  Ⅎwnfc 2738  ∃*wrmo 2899   ⊆ wss 3540  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-in 3547  df-ss 3554  df-disj 4554 This theorem is referenced by:  disjeq1f  28769  esumrnmpt2  29457  measvuni  29604
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