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Theorem 0disj 3926
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 3401 . . 3  |-  (/)  C_  { x }
21rgenw 2487 . 2  |-  A. x  e.  A  (/)  C_  { x }
3 sndisj 3925 . 2  |- Disj  x  e.  A  { x }
4 disjss2 3909 . 2  |-  ( A. x  e.  A  (/)  C_  { x }  ->  (Disj  x  e.  A  { x }  -> Disj  x  e.  A  (/) ) )
52, 3, 4mp2 16 1  |- Disj  x  e.  A  (/)
Colors of variables: wff set class
Syntax hints:   A.wral 2416    C_ wss 3071   (/)c0 3363   {csn 3527  Disj wdisj 3906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rmo 2424  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533  df-disj 3907
This theorem is referenced by: (None)
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