MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0disj Structured version   Unicode version

Theorem 0disj 4419
Description: Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
0disj  |- Disj  x  e.  A  (/)

Proof of Theorem 0disj
StepHypRef Expression
1 0ss 3797 . . 3  |-  (/)  C_  { x }
21rgenw 2793 . 2  |-  A. x  e.  A  (/)  C_  { x }
3 sndisj 4418 . 2  |- Disj  x  e.  A  { x }
4 disjss2 4400 . 2  |-  ( A. x  e.  A  (/)  C_  { x }  ->  (Disj  x  e.  A  { x }  -> Disj  x  e.  A  (/) ) )
52, 3, 4mp2 9 1  |- Disj  x  e.  A  (/)
Colors of variables: wff setvar class
Syntax hints:   A.wral 2782    C_ wss 3442   (/)c0 3767   {csn 4002  Disj wdisj 4397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rmo 2790  df-v 3089  df-dif 3445  df-in 3449  df-ss 3456  df-nul 3768  df-sn 4003  df-disj 4398
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator