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Theorem unidif0 4374
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 4036 . . . 4  |-  U. (
( A  \  { (/)
} )  u.  { (/)
} )  =  ( U. ( A  \  { (/) } )  u. 
U. { (/) } )
2 undif1 3705 . . . . . 6  |-  ( ( A  \  { (/) } )  u.  { (/) } )  =  ( A  u.  { (/) } )
3 uncom 3493 . . . . . 6  |-  ( A  u.  { (/) } )  =  ( { (/) }  u.  A )
42, 3eqtr2i 2459 . . . . 5  |-  ( {
(/) }  u.  A
)  =  ( ( A  \  { (/) } )  u.  { (/) } )
54unieqi 4027 . . . 4  |-  U. ( { (/) }  u.  A
)  =  U. (
( A  \  { (/)
} )  u.  { (/)
} )
6 0ex 4341 . . . . . . 7  |-  (/)  e.  _V
76unisn 4033 . . . . . 6  |-  U. { (/)
}  =  (/)
87uneq2i 3500 . . . . 5  |-  ( U. ( A  \  { (/) } )  u.  U. { (/)
} )  =  ( U. ( A  \  { (/) } )  u.  (/) )
9 un0 3654 . . . . 5  |-  ( U. ( A  \  { (/) } )  u.  (/) )  = 
U. ( A  \  { (/) } )
108, 9eqtr2i 2459 . . . 4  |-  U. ( A  \  { (/) } )  =  ( U. ( A  \  { (/) } )  u.  U. { (/) } )
111, 5, 103eqtr4ri 2469 . . 3  |-  U. ( A  \  { (/) } )  =  U. ( {
(/) }  u.  A
)
12 uniun 4036 . . 3  |-  U. ( { (/) }  u.  A
)  =  ( U. { (/) }  u.  U. A )
137uneq1i 3499 . . 3  |-  ( U. { (/) }  u.  U. A )  =  (
(/)  u.  U. A )
1411, 12, 133eqtri 2462 . 2  |-  U. ( A  \  { (/) } )  =  ( (/)  u.  U. A )
15 uncom 3493 . 2  |-  ( (/)  u. 
U. A )  =  ( U. A  u.  (/) )
16 un0 3654 . 2  |-  ( U. A  u.  (/) )  = 
U. A
1714, 15, 163eqtri 2462 1  |-  U. ( A  \  { (/) } )  =  U. A
Colors of variables: wff set class
Syntax hints:    = wceq 1653    \ cdif 3319    u. cun 3320   (/)c0 3630   {csn 3816   U.cuni 4017
This theorem is referenced by:  infeq5i  7593  zornn0g  8387  basdif0  17020  tgdif0  17059  stoweidlem57  27784
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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  ax-nul 4340
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-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-sn 3822  df-pr 3823  df-uni 4018
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