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Theorem disjss1f 30250
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 4029 . . 3 (𝐴𝐵 → (∃*𝑥𝐵 𝑦𝐶 → ∃*𝑥𝐴 𝑦𝐶))
43alimdv 1908 . 2 (𝐴𝐵 → (∀𝑦∃*𝑥𝐵 𝑦𝐶 → ∀𝑦∃*𝑥𝐴 𝑦𝐶))
5 df-disj 5023 . 2 (Disj 𝑥𝐵 𝐶 ↔ ∀𝑦∃*𝑥𝐵 𝑦𝐶)
6 df-disj 5023 . 2 (Disj 𝑥𝐴 𝐶 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐶)
74, 5, 63imtr4g 297 1 (𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1526  wcel 2105  wnfc 2958  ∃*wrmo 3138  wss 3933  Disj wdisj 5022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rmo 3143  df-in 3940  df-ss 3949  df-disj 5023
This theorem is referenced by:  disjeq1f  30251  esumrnmpt2  31226  measvuni  31372
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