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Theorem untint 24942
Description: If there is an untangled element of a class, then the intersection of the class is untangled. (Contributed by Scott Fenton, 1-Mar-2011.)
Assertion
Ref Expression
untint  |-  ( E. x  e.  A  A. y  e.  x  -.  y  e.  y  ->  A. y  e.  |^| A  -.  y  e.  y
)
Distinct variable group:    x, y, A

Proof of Theorem untint
StepHypRef Expression
1 intss1 4009 . . 3  |-  ( x  e.  A  ->  |^| A  C_  x )
2 ssralv 3352 . . 3  |-  ( |^| A  C_  x  ->  ( A. y  e.  x  -.  y  e.  y  ->  A. y  e.  |^| A  -.  y  e.  y ) )
31, 2syl 16 . 2  |-  ( x  e.  A  ->  ( A. y  e.  x  -.  y  e.  y  ->  A. y  e.  |^| A  -.  y  e.  y ) )
43rexlimiv 2769 1  |-  ( E. x  e.  A  A. y  e.  x  -.  y  e.  y  ->  A. y  e.  |^| A  -.  y  e.  y
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1717   A.wral 2651   E.wrex 2652    C_ wss 3265   |^|cint 3994
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ral 2656  df-rex 2657  df-v 2903  df-in 3272  df-ss 3279  df-int 3995
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