MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  disjss2 Structured version   Visualization version   GIF version

Theorem disjss2 4815
Description: If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjss2 (∀𝑥𝐴 𝐵𝐶 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐴 𝐵))

Proof of Theorem disjss2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssel 3792 . . . . 5 (𝐵𝐶 → (𝑦𝐵𝑦𝐶))
21ralimi 3140 . . . 4 (∀𝑥𝐴 𝐵𝐶 → ∀𝑥𝐴 (𝑦𝐵𝑦𝐶))
3 rmoim 3605 . . . 4 (∀𝑥𝐴 (𝑦𝐵𝑦𝐶) → (∃*𝑥𝐴 𝑦𝐶 → ∃*𝑥𝐴 𝑦𝐵))
42, 3syl 17 . . 3 (∀𝑥𝐴 𝐵𝐶 → (∃*𝑥𝐴 𝑦𝐶 → ∃*𝑥𝐴 𝑦𝐵))
54alimdv 2007 . 2 (∀𝑥𝐴 𝐵𝐶 → (∀𝑦∃*𝑥𝐴 𝑦𝐶 → ∀𝑦∃*𝑥𝐴 𝑦𝐵))
6 df-disj 4813 . 2 (Disj 𝑥𝐴 𝐶 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐶)
7 df-disj 4813 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐵)
85, 6, 73imtr4g 287 1 (∀𝑥𝐴 𝐵𝐶 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐴 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1635  wcel 2156  wral 3096  ∃*wrmo 3099  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-ral 3101  df-rmo 3104  df-in 3776  df-ss 3783  df-disj 4813
This theorem is referenced by:  disjeq2  4816  0disj  4837  uniioombllem2  23564  uniioombllem4  23567  disjxwwlksn  27041  disjxwwlkn  27051  fusgreghash2wspv  27510  fsumiunss  40287
  Copyright terms: Public domain W3C validator