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Theorem trintss 4318
Description: If  A is transitive and non-null, then  |^| A is a subset of  A. (Contributed by Scott Fenton, 3-Mar-2011.)
Assertion
Ref Expression
trintss  |-  ( ( A  =/=  (/)  /\  Tr  A )  ->  |^| A  C_  A )

Proof of Theorem trintss
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2959 . . . 4  |-  y  e. 
_V
21elint2 4057 . . 3  |-  ( y  e.  |^| A  <->  A. x  e.  A  y  e.  x )
3 r19.2z 3717 . . . . 5  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  y  e.  x )  ->  E. x  e.  A  y  e.  x )
43ex 424 . . . 4  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  y  e.  x  ->  E. x  e.  A  y  e.  x ) )
5 trel 4309 . . . . . 6  |-  ( Tr  A  ->  ( (
y  e.  x  /\  x  e.  A )  ->  y  e.  A ) )
65exp3acom23 1381 . . . . 5  |-  ( Tr  A  ->  ( x  e.  A  ->  ( y  e.  x  ->  y  e.  A ) ) )
76rexlimdv 2829 . . . 4  |-  ( Tr  A  ->  ( E. x  e.  A  y  e.  x  ->  y  e.  A ) )
84, 7sylan9 639 . . 3  |-  ( ( A  =/=  (/)  /\  Tr  A )  ->  ( A. x  e.  A  y  e.  x  ->  y  e.  A ) )
92, 8syl5bi 209 . 2  |-  ( ( A  =/=  (/)  /\  Tr  A )  ->  (
y  e.  |^| A  ->  y  e.  A ) )
109ssrdv 3354 1  |-  ( ( A  =/=  (/)  /\  Tr  A )  ->  |^| A  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706    C_ wss 3320   (/)c0 3628   |^|cint 4050   Tr wtr 4302
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-ne 2601  df-ral 2710  df-rex 2711  df-v 2958  df-dif 3323  df-in 3327  df-ss 3334  df-nul 3629  df-uni 4016  df-int 4051  df-tr 4303
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