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Theorem 0disj 4016
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 3483 . . 3  |-  (/)  C_  { x }
21rgenw 2610 . 2  |-  A. x  e.  A  (/)  C_  { x }
3 sndisj 4015 . 2  |- Disj  x  e.  A { x }
4 disjss2 3996 . 2  |-  ( A. x  e.  A  (/)  C_  { x }  ->  (Disj  x  e.  A { x }  -> Disj  x  e.  A (/) ) )
52, 3, 4mp2 17 1  |- Disj  x  e.  A (/)
Colors of variables: wff set class
Syntax hints:   A.wral 2543    C_ wss 3152   (/)c0 3455   {csn 3640  Disj wdisj 3993
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rmo 2551  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-sn 3646  df-disj 3994
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