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Theorem unidif 3828
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 3827 . . 3  |-  ( A. x  e.  A  E. y  e.  ( A  \  B ) x  C_  y  ->  U. A  C_  U. ( A  \  B ) )
2 difss 3253 . . . 4  |-  ( A 
\  B )  C_  A
32unissi 3819 . . 3  |-  U. ( A  \  B )  C_  U. A
41, 3jctil 310 . 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 3162 . 2  |-  ( U. ( A  \  B )  =  U. A  <->  ( U. ( A  \  B ) 
C_  U. A  /\  U. A  C_  U. ( A 
\  B ) ) )
64, 5sylibr 133 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 103    = wceq 1348   A.wral 2448   E.wrex 2449    \ cdif 3118    C_ wss 3121   U.cuni 3796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-uni 3797
This theorem is referenced by: (None)
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