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Theorem 0disj 4440
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 3814 . . 3  |-  (/)  C_  { x }
21rgenw 2825 . 2  |-  A. x  e.  A  (/)  C_  { x }
3 sndisj 4439 . 2  |- Disj  x  e.  A  { x }
4 disjss2 4420 . 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 2814    C_ wss 3476   (/)c0 3785   {csn 4027  Disj wdisj 4417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rmo 2822  df-v 3115  df-dif 3479  df-in 3483  df-ss 3490  df-nul 3786  df-sn 4028  df-disj 4418
This theorem is referenced by: (None)
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