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Theorem 0disj 4173
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 3624 . . 3  |-  (/)  C_  { x }
21rgenw 2741 . 2  |-  A. x  e.  A  (/)  C_  { x }
3 sndisj 4172 . 2  |- Disj  x  e.  A { x }
4 disjss2 4153 . 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 set class
Syntax hints:   A.wral 2674    C_ wss 3288   (/)c0 3596   {csn 3782  Disj wdisj 4150
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rmo 2682  df-v 2926  df-dif 3291  df-in 3295  df-ss 3302  df-nul 3597  df-sn 3788  df-disj 4151
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