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Theorem undif4 3281
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 pm2.621 666 . . . . . . 7  C  C  C
2 olc 632 . . . . . . 7  C  C
31, 2impbid1 130 . . . . . 6  C  C  C
43anbi2d 437 . . . . 5  C  C  C
5 eldif 2924 . . . . . . 7  \  C  C
65orbi2i 679 . . . . . 6  \  C  C
7 ordi 729 . . . . . 6  C  C
86, 7bitri 173 . . . . 5  \  C  C
9 elun 3081 . . . . . 6  u.
109anbi1i 431 . . . . 5  u.  C  C
114, 8, 103bitr4g 212 . . . 4  C  \  C  u.  C
12 elun 3081 . . . 4  u.  \  C  \  C
13 eldif 2924 . . . 4  u.  \  C  u.  C
1411, 12, 133bitr4g 212 . . 3  C  u. 
\  C  u. 
\  C
1514alimi 1344 . 2  C  u.  \  C  u.  \  C
16 disj1 3267 . 2  i^i  C  (/)  C
17 dfcleq 2034 . 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 629  wal 1241   wceq 1243   wcel 1393    \ cdif 2911    u. cun 2912    i^i cin 2913   (/)c0 3221
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 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-v 2556  df-dif 2917  df-un 2919  df-in 2921  df-nul 3222
This theorem is referenced by: (None)
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