ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  disjss2 GIF version

Theorem disjss2 3909
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 3091 . . . . 5 (𝐵𝐶 → (𝑦𝐵𝑦𝐶))
21ralimi 2495 . . . 4 (∀𝑥𝐴 𝐵𝐶 → ∀𝑥𝐴 (𝑦𝐵𝑦𝐶))
3 rmoim 2885 . . . 4 (∀𝑥𝐴 (𝑦𝐵𝑦𝐶) → (∃*𝑥𝐴 𝑦𝐶 → ∃*𝑥𝐴 𝑦𝐵))
42, 3syl 14 . . 3 (∀𝑥𝐴 𝐵𝐶 → (∃*𝑥𝐴 𝑦𝐶 → ∃*𝑥𝐴 𝑦𝐵))
54alimdv 1851 . 2 (∀𝑥𝐴 𝐵𝐶 → (∀𝑦∃*𝑥𝐴 𝑦𝐶 → ∀𝑦∃*𝑥𝐴 𝑦𝐵))
6 df-disj 3907 . 2 (Disj 𝑥𝐴 𝐶 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐶)
7 df-disj 3907 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐵)
85, 6, 73imtr4g 204 1 (∀𝑥𝐴 𝐵𝐶 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐴 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1329  wcel 1480  wral 2416  ∃*wrmo 2419  wss 3071  Disj wdisj 3906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-ral 2421  df-rmo 2424  df-in 3077  df-ss 3084  df-disj 3907
This theorem is referenced by:  disjeq2  3910  0disj  3926
  Copyright terms: Public domain W3C validator