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Theorem unidif0 3883
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.  \  { (/) }  U.

Proof of Theorem unidif0
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0i 3197 . . . . . . 7  (/)
21pm4.71i 371 . . . . . 6  (/)
32anbi1i 431 . . . . 5  (/)
4 an32 481 . . . . 5  (/)  (/)
5 anass 381 . . . . 5  (/)  (/)
63, 4, 53bitr2ri 198 . . . 4  (/)
76exbii 1469 . . 3  (/)
8 eluni 3546 . . . 4  U. 
\  { (/) }  \  { (/)
}
9 eldif 2895 . . . . . . 7  \  { (/) }  { (/) }
10 elsn 3354 . . . . . . . . 9  { (/) }  (/)
1110notbii 578 . . . . . . . 8  { (/) }  (/)
1211anbi2i 430 . . . . . . 7  { (/) }  (/)
139, 12bitri 173 . . . . . 6  \  { (/) }  (/)
1413anbi2i 430 . . . . 5  \  { (/) }  (/)
1514exbii 1469 . . . 4  \  { (/) }  (/)
168, 15bitri 173 . . 3  U. 
\  { (/) }  (/)
17 eluni 3546 . . 3  U.
187, 16, 173bitr4i 201 . 2  U. 
\  { (/) }  U.
1918eqriv 2010 1  U.  \  { (/) }  U.
Colors of variables: wff set class
Syntax hints:   wn 3   wa 97   wceq 1223  wex 1354   wcel 1366    \ cdif 2882   (/)c0 3192   {csn 3339   U.cuni 3543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 529  ax-in2 530  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995
This theorem depends on definitions:  df-bi 110  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-v 2528  df-dif 2888  df-nul 3193  df-sn 3345  df-uni 3544
This theorem is referenced by: (None)
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