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Theorem iunxdif2 3675
Description: Indexed union with a class difference as its index. (Contributed by NM, 10-Dec-2004.)
Hypothesis
Ref Expression
iunxdif2.1  C  D
Assertion
Ref Expression
iunxdif2  \  C  C_  D  U_  \  D  U_  C
Distinct variable groups:   ,,   ,,   , C   , D
Allowed substitution hints:    C()    D()

Proof of Theorem iunxdif2
StepHypRef Expression
1 iunss2 3672 . . 3  \  C  C_  D  U_  C  C_  U_  \  D
2 difss 3043 . . . . 5 
\  C_
3 iunss1 3638 . . . . 5  \ 
C_  U_  \  D  C_  U_  D
42, 3ax-mp 7 . . . 4  U_  \  D  C_  U_  D
5 iunxdif2.1 . . . . 5  C  D
65cbviunv 3666 . . . 4  U_  C  U_  D
74, 6sseqtr4i 2951 . . 3  U_  \  D  C_  U_  C
81, 7jctil 295 . 2  \  C  C_  D  U_  \  D  C_  U_  C  U_  C  C_ 
U_  \  D
9 eqss 2933 . 2  U_  \  D 
U_  C  U_  \  D  C_  U_  C  U_  C  C_ 
U_  \  D
108, 9sylibr 137 1  \  C  C_  D  U_  \  D  U_  C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1226  wral 2280  wrex 2281    \ cdif 2887    C_ wss 2890   U_ciun 3627
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-dif 2893  df-in 2897  df-ss 2904  df-iun 3629
This theorem is referenced by: (None)
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