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Theorem 0disj 4208
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 3658 . . 3  |-  (/)  C_  { x }
21rgenw 2775 . 2  |-  A. x  e.  A  (/)  C_  { x }
3 sndisj 4207 . 2  |- Disj  x  e.  A { x }
4 disjss2 4188 . 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 2707    C_ wss 3322   (/)c0 3630   {csn 3816  Disj wdisj 4185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rmo 2715  df-v 2960  df-dif 3325  df-in 3329  df-ss 3336  df-nul 3631  df-sn 3822  df-disj 4186
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