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Theorem unidif 3859
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 3858 . . 3  |-  ( A. x  e.  A  E. y  e.  ( A  \  B ) x  C_  y  ->  U. A  C_  U. ( A  \  B ) )
2 difss 3303 . . . 4  |-  ( A 
\  B )  C_  A
3 uniss 3848 . . . 4  |-  ( ( A  \  B ) 
C_  A  ->  U. ( A  \  B )  C_  U. A )
42, 3ax-mp 8 . . 3  |-  U. ( A  \  B )  C_  U. A
51, 4jctil 523 . 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 ) ) )
6 eqss 3194 . 2  |-  ( U. ( A  \  B )  =  U. A  <->  ( U. ( A  \  B ) 
C_  U. A  /\  U. A  C_  U. ( A 
\  B ) ) )
75, 6sylibr 203 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 358    = wceq 1623   A.wral 2543   E.wrex 2544    \ cdif 3149    C_ wss 3152   U.cuni 3827
This theorem is referenced by:  ordunidif  4440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-uni 3828
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