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Theorem untint 29789
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 4239 . . 3  |-  ( x  e.  A  ->  |^| A  C_  x )
2 ssralv 3500 . . 3  |-  ( |^| A  C_  x  ->  ( A. y  e.  x  -.  y  e.  y  ->  A. y  e.  |^| A  -.  y  e.  y ) )
31, 2syl 17 . 2  |-  ( x  e.  A  ->  ( A. y  e.  x  -.  y  e.  y  ->  A. y  e.  |^| A  -.  y  e.  y ) )
43rexlimiv 2887 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 1840   A.wral 2751   E.wrex 2752    C_ wss 3411   |^|cint 4224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ral 2756  df-rex 2757  df-v 3058  df-in 3418  df-ss 3425  df-int 4225
This theorem is referenced by: (None)
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