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Theorem 0disj 3986
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 3453 . . 3  |-  (/)  C_  { x }
21rgenw 2525 . 2  |-  A. x  e.  A  (/)  C_  { x }
3 sndisj 3985 . 2  |- Disj  x  e.  A  { x }
4 disjss2 3969 . 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 2448    C_ wss 3121   (/)c0 3414   {csn 3583  Disj wdisj 3966
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rmo 2456  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3589  df-disj 3967
This theorem is referenced by: (None)
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