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Theorem symdif0 24753
Description: Symmetric difference with the empty class. (Contributed by Scott Fenton, 24-Apr-2012.)
Assertion
Ref Expression
symdif0  |-  ( A(++) (/) )  =  A

Proof of Theorem symdif0
StepHypRef Expression
1 df-symdif 24747 . 2  |-  ( A(++) (/) )  =  (
( A  \  (/) )  u.  ( (/)  \  A ) )
2 dif0 3558 . . 3  |-  ( A 
\  (/) )  =  A
3 0dif 3559 . . 3  |-  ( (/)  \  A )  =  (/)
42, 3uneq12i 3361 . 2  |-  ( ( A  \  (/) )  u.  ( (/)  \  A ) )  =  ( A  u.  (/) )
5 un0 3513 . 2  |-  ( A  u.  (/) )  =  A
61, 4, 53eqtri 2340 1  |-  ( A(++) (/) )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1633    \ cdif 3183    u. cun 3184   (/)c0 3489  (++)csymdif 24746
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-symdif 24747
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