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Theorem disjss1 4818
Description: A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjss1 (𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem disjss1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssel 3792 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21anim1d 600 . . . 4 (𝐴𝐵 → ((𝑥𝐴𝑦𝐶) → (𝑥𝐵𝑦𝐶)))
32moimd 2684 . . 3 (𝐴𝐵 → (∃*𝑥(𝑥𝐵𝑦𝐶) → ∃*𝑥(𝑥𝐴𝑦𝐶)))
43alimdv 2007 . 2 (𝐴𝐵 → (∀𝑦∃*𝑥(𝑥𝐵𝑦𝐶) → ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐶)))
5 dfdisj2 4814 . 2 (Disj 𝑥𝐵 𝐶 ↔ ∀𝑦∃*𝑥(𝑥𝐵𝑦𝐶))
6 dfdisj2 4814 . 2 (Disj 𝑥𝐴 𝐶 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐶))
74, 5, 63imtr4g 287 1 (𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1635  wcel 2156  ∃*wmo 2631  wss 3769  Disj wdisj 4812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-rmo 3104  df-in 3776  df-ss 3783  df-disj 4813
This theorem is referenced by:  disjeq1  4819  disjx0  4839  disjxiun  4841  disjss3  4843  volfiniun  23528  uniioovol  23560  uniioombllem4  23567  disjiunel  29734  carsggect  30705  carsgclctunlem2  30706  omsmeas  30710  sibfof  30727  disjf1o  39867  fsumiunss  40287  sge0iunmptlemre  41111  meadjiunlem  41161  meaiuninclem  41176
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