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Theorem disjss1f 29228
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 29180 . . 3 (𝐴𝐵 → (∃*𝑥𝐵 𝑦𝐶 → ∃*𝑥𝐴 𝑦𝐶))
43alimdv 1842 . 2 (𝐴𝐵 → (∀𝑦∃*𝑥𝐵 𝑦𝐶 → ∀𝑦∃*𝑥𝐴 𝑦𝐶))
5 df-disj 4584 . 2 (Disj 𝑥𝐵 𝐶 ↔ ∀𝑦∃*𝑥𝐵 𝑦𝐶)
6 df-disj 4584 . 2 (Disj 𝑥𝐴 𝐶 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐶)
74, 5, 63imtr4g 285 1 (𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478  wcel 1987  wnfc 2748  ∃*wrmo 2910  wss 3555  Disj wdisj 4583
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-in 3562  df-ss 3569  df-disj 4584
This theorem is referenced by:  disjeq1f  29229  esumrnmpt2  29908  measvuni  30055
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