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Theorem trintss 4166
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 2825 . . . 4  |-  y  e. 
_V
21elint2 3906 . . 3  |-  ( y  e.  |^| A  <->  A. x  e.  A  y  e.  x )
3 r19.2z 3577 . . . . 5  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  y  e.  x )  ->  E. x  e.  A  y  e.  x )
43ex 423 . . . 4  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  y  e.  x  ->  E. x  e.  A  y  e.  x ) )
5 trel 4157 . . . . . 6  |-  ( Tr  A  ->  ( (
y  e.  x  /\  x  e.  A )  ->  y  e.  A ) )
65exp3acom23 1363 . . . . 5  |-  ( Tr  A  ->  ( x  e.  A  ->  ( y  e.  x  ->  y  e.  A ) ) )
76rexlimdv 2700 . . . 4  |-  ( Tr  A  ->  ( E. x  e.  A  y  e.  x  ->  y  e.  A ) )
84, 7sylan9 638 . . 3  |-  ( ( A  =/=  (/)  /\  Tr  A )  ->  ( A. x  e.  A  y  e.  x  ->  y  e.  A ) )
92, 8syl5bi 208 . 2  |-  ( ( A  =/=  (/)  /\  Tr  A )  ->  (
y  e.  |^| A  ->  y  e.  A ) )
109ssrdv 3219 1  |-  ( ( A  =/=  (/)  /\  Tr  A )  ->  |^| A  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1701    =/= wne 2479   A.wral 2577   E.wrex 2578    C_ wss 3186   (/)c0 3489   |^|cint 3899   Tr wtr 4150
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-v 2824  df-dif 3189  df-in 3193  df-ss 3200  df-nul 3490  df-uni 3865  df-int 3900  df-tr 4151
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