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Theorem undif4 3261
Description: Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
undif4  i^i  C  (/)  u.  \  C  u.  \  C

Proof of Theorem undif4
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 pm2.621 653 . . . . . . 7  C  C  C
2 olc 619 . . . . . . 7  C  C
31, 2impbid1 130 . . . . . 6  C  C  C
43anbi2d 440 . . . . 5  C  C  C
5 eldif 2904 . . . . . . 7  \  C  C
65orbi2i 666 . . . . . 6  \  C  C
7 ordi 717 . . . . . 6  C  C
86, 7bitri 173 . . . . 5  \  C  C
9 elun 3061 . . . . . 6  u.
109anbi1i 434 . . . . 5  u.  C  C
114, 8, 103bitr4g 212 . . . 4  C  \  C  u.  C
12 elun 3061 . . . 4  u.  \  C  \  C
13 eldif 2904 . . . 4  u.  \  C  u.  C
1411, 12, 133bitr4g 212 . . 3  C  u. 
\  C  u. 
\  C
1514alimi 1324 . 2  C  u.  \  C  u.  \  C
16 disj1 3247 . 2  i^i  C  (/)  C
17 dfcleq 2016 . 2  u. 
\  C  u.  \  C  u.  \  C  u.  \  C
1815, 16, 173imtr4i 190 1  i^i  C  (/)  u.  \  C  u.  \  C
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wb 98   wo 616  wal 1226   wceq 1228   wcel 1374    \ cdif 2891    u. cun 2892    i^i cin 2893   (/)c0 3201
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-nul 3202
This theorem is referenced by: (None)
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