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Theorem untint 24058
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 3877 . . 3  |-  ( x  e.  A  ->  |^| A  C_  x )
2 ssralv 3237 . . 3  |-  ( |^| A  C_  x  ->  ( A. y  e.  x  -.  y  e.  y  ->  A. y  e.  |^| A  -.  y  e.  y ) )
31, 2syl 15 . 2  |-  ( x  e.  A  ->  ( A. y  e.  x  -.  y  e.  y  ->  A. y  e.  |^| A  -.  y  e.  y ) )
43rexlimiv 2661 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 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   |^|cint 3862
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-in 3159  df-ss 3166  df-int 3863
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