Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  disjss1f Structured version   Visualization version   GIF version

Theorem disjss1f 29705
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 29654 . . 3 (𝐴𝐵 → (∃*𝑥𝐵 𝑦𝐶 → ∃*𝑥𝐴 𝑦𝐶))
43alimdv 2007 . 2 (𝐴𝐵 → (∀𝑦∃*𝑥𝐵 𝑦𝐶 → ∀𝑦∃*𝑥𝐴 𝑦𝐶))
5 df-disj 4806 . 2 (Disj 𝑥𝐵 𝐶 ↔ ∀𝑦∃*𝑥𝐵 𝑦𝐶)
6 df-disj 4806 . 2 (Disj 𝑥𝐴 𝐶 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐶)
74, 5, 63imtr4g 287 1 (𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1635  wcel 2155  wnfc 2931  ∃*wrmo 3095  wss 3763  Disj wdisj 4805
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 2067  ax-7 2103  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781
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 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-rmo 3100  df-in 3770  df-ss 3777  df-disj 4806
This theorem is referenced by:  disjeq1f  29706  esumrnmpt2  30449  measvuni  30596
  Copyright terms: Public domain W3C validator