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Theorem untint 28547
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 4292 . . 3  |-  ( x  e.  A  ->  |^| A  C_  x )
2 ssralv 3559 . . 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 2944 1  |-  ( E. x  e.  A  A. y  e.  x  -.  y  e.  y  ->  A. y  e.  |^| A  -.  y  e.  y
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1762   A.wral 2809   E.wrex 2810    C_ wss 3471   |^|cint 4277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ral 2814  df-rex 2815  df-v 3110  df-in 3478  df-ss 3485  df-int 4278
This theorem is referenced by: (None)
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