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Theorem untint 25161
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 4065 . . 3  |-  ( x  e.  A  ->  |^| A  C_  x )
2 ssralv 3407 . . 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 2824 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 1725   A.wral 2705   E.wrex 2706    C_ wss 3320   |^|cint 4050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-v 2958  df-in 3327  df-ss 3334  df-int 4051
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