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Theorem unidif 4196
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 4195 . . 3  |-  ( A. x  e.  A  E. y  e.  ( A  \  B ) x  C_  y  ->  U. A  C_  U. ( A  \  B ) )
2 difss 3545 . . . 4  |-  ( A 
\  B )  C_  A
32unissi 4186 . . 3  |-  U. ( A  \  B )  C_  U. A
41, 3jctil 535 . 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 3432 . 2  |-  ( U. ( A  \  B )  =  U. A  <->  ( U. ( A  \  B ) 
C_  U. A  /\  U. A  C_  U. ( A 
\  B ) ) )
64, 5sylibr 212 1  |-  ( A. x  e.  A  E. y  e.  ( A  \  B ) x  C_  y  ->  U. ( A  \  B )  =  U. A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399   A.wral 2732   E.wrex 2733    \ cdif 3386    C_ wss 3389   U.cuni 4163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ral 2737  df-rex 2738  df-v 3036  df-dif 3392  df-in 3396  df-ss 3403  df-uni 4164
This theorem is referenced by:  ordunidif  4840
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