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Theorem trintss 4513
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 3048 . . . 4  |-  y  e. 
_V
21elint2 4241 . . 3  |-  ( y  e.  |^| A  <->  A. x  e.  A  y  e.  x )
3 r19.2z 3858 . . . . 5  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  y  e.  x )  ->  E. x  e.  A  y  e.  x )
43ex 436 . . . 4  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  y  e.  x  ->  E. x  e.  A  y  e.  x ) )
5 trel 4504 . . . . . 6  |-  ( Tr  A  ->  ( (
y  e.  x  /\  x  e.  A )  ->  y  e.  A ) )
65expcomd 440 . . . . 5  |-  ( Tr  A  ->  ( x  e.  A  ->  ( y  e.  x  ->  y  e.  A ) ) )
76rexlimdv 2877 . . . 4  |-  ( Tr  A  ->  ( E. x  e.  A  y  e.  x  ->  y  e.  A ) )
84, 7sylan9 663 . . 3  |-  ( ( A  =/=  (/)  /\  Tr  A )  ->  ( A. x  e.  A  y  e.  x  ->  y  e.  A ) )
92, 8syl5bi 221 . 2  |-  ( ( A  =/=  (/)  /\  Tr  A )  ->  (
y  e.  |^| A  ->  y  e.  A ) )
109ssrdv 3438 1  |-  ( ( A  =/=  (/)  /\  Tr  A )  ->  |^| A  C_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    e. wcel 1887    =/= wne 2622   A.wral 2737   E.wrex 2738    C_ wss 3404   (/)c0 3731   |^|cint 4234   Tr wtr 4497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-v 3047  df-dif 3407  df-in 3411  df-ss 3418  df-nul 3732  df-uni 4199  df-int 4235  df-tr 4498
This theorem is referenced by: (None)
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