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Theorem unidif0 4574
Description: The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)
Assertion
Ref Expression
unidif0  |-  U. ( A  \  { (/) } )  =  U. A

Proof of Theorem unidif0
StepHypRef Expression
1 uniun 4209 . . . 4  |-  U. (
( A  \  { (/)
} )  u.  { (/)
} )  =  ( U. ( A  \  { (/) } )  u. 
U. { (/) } )
2 undif1 3833 . . . . . 6  |-  ( ( A  \  { (/) } )  u.  { (/) } )  =  ( A  u.  { (/) } )
3 uncom 3569 . . . . . 6  |-  ( A  u.  { (/) } )  =  ( { (/) }  u.  A )
42, 3eqtr2i 2494 . . . . 5  |-  ( {
(/) }  u.  A
)  =  ( ( A  \  { (/) } )  u.  { (/) } )
54unieqi 4199 . . . 4  |-  U. ( { (/) }  u.  A
)  =  U. (
( A  \  { (/)
} )  u.  { (/)
} )
6 0ex 4528 . . . . . . 7  |-  (/)  e.  _V
76unisn 4205 . . . . . 6  |-  U. { (/)
}  =  (/)
87uneq2i 3576 . . . . 5  |-  ( U. ( A  \  { (/) } )  u.  U. { (/)
} )  =  ( U. ( A  \  { (/) } )  u.  (/) )
9 un0 3762 . . . . 5  |-  ( U. ( A  \  { (/) } )  u.  (/) )  = 
U. ( A  \  { (/) } )
108, 9eqtr2i 2494 . . . 4  |-  U. ( A  \  { (/) } )  =  ( U. ( A  \  { (/) } )  u.  U. { (/) } )
111, 5, 103eqtr4ri 2504 . . 3  |-  U. ( A  \  { (/) } )  =  U. ( {
(/) }  u.  A
)
12 uniun 4209 . . 3  |-  U. ( { (/) }  u.  A
)  =  ( U. { (/) }  u.  U. A )
137uneq1i 3575 . . 3  |-  ( U. { (/) }  u.  U. A )  =  (
(/)  u.  U. A )
1411, 12, 133eqtri 2497 . 2  |-  U. ( A  \  { (/) } )  =  ( (/)  u.  U. A )
15 uncom 3569 . 2  |-  ( (/)  u. 
U. A )  =  ( U. A  u.  (/) )
16 un0 3762 . 2  |-  ( U. A  u.  (/) )  = 
U. A
1714, 15, 163eqtri 2497 1  |-  U. ( A  \  { (/) } )  =  U. A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1452    \ cdif 3387    u. cun 3388   (/)c0 3722   {csn 3959   U.cuni 4190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-nul 4527
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-sn 3960  df-pr 3962  df-uni 4191
This theorem is referenced by:  infeq5i  8159  zornn0g  8953  basdif0  20045  tgdif0  20085  omsmeas  29228  omsmeasOLD  29229  stoweidlem57  38030
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