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Theorem disjss1f 32574
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, 2ssrmof 4050 . . 3 (𝐴𝐵 → (∃*𝑥𝐵 𝑦𝐶 → ∃*𝑥𝐴 𝑦𝐶))
43alimdv 1916 . 2 (𝐴𝐵 → (∀𝑦∃*𝑥𝐵 𝑦𝐶 → ∀𝑦∃*𝑥𝐴 𝑦𝐶))
5 df-disj 5109 . 2 (Disj 𝑥𝐵 𝐶 ↔ ∀𝑦∃*𝑥𝐵 𝑦𝐶)
6 df-disj 5109 . 2 (Disj 𝑥𝐴 𝐶 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐶)
74, 5, 63imtr4g 296 1 (𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1538  wcel 2108  wnfc 2889  ∃*wrmo 3378  wss 3950  Disj wdisj 5108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-11 2157  ax-12 2177
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-mo 2539  df-clel 2815  df-nfc 2891  df-rmo 3379  df-ss 3967  df-disj 5109
This theorem is referenced by:  disjeq1f  32575  esumrnmpt2  34047  measvuni  34193
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