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Theorem unidif 4039
Description: If the difference  A  \  B contains the largest members of  A, then the union of the difference is the union of  A. (Contributed by NM, 22-Mar-2004.)
Assertion
Ref Expression
unidif  |-  ( A. x  e.  A  E. y  e.  ( A  \  B ) x  C_  y  ->  U. ( A  \  B )  =  U. A )
Distinct variable groups:    x, y, A    x, B, y

Proof of Theorem unidif
StepHypRef Expression
1 uniss2 4038 . . 3  |-  ( A. x  e.  A  E. y  e.  ( A  \  B ) x  C_  y  ->  U. A  C_  U. ( A  \  B ) )
2 difss 3466 . . . 4  |-  ( A 
\  B )  C_  A
32unissi 4030 . . 3  |-  U. ( A  \  B )  C_  U. A
41, 3jctil 524 . 2  |-  ( A. x  e.  A  E. y  e.  ( A  \  B ) x  C_  y  ->  ( U. ( A  \  B )  C_  U. A  /\  U. A  C_ 
U. ( A  \  B ) ) )
5 eqss 3355 . 2  |-  ( U. ( A  \  B )  =  U. A  <->  ( U. ( A  \  B ) 
C_  U. A  /\  U. A  C_  U. ( A 
\  B ) ) )
64, 5sylibr 204 1  |-  ( A. x  e.  A  E. y  e.  ( A  \  B ) x  C_  y  ->  U. ( A  \  B )  =  U. A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652   A.wral 2697   E.wrex 2698    \ cdif 3309    C_ wss 3312   U.cuni 4007
This theorem is referenced by:  ordunidif  4621
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-uni 4008
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