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Theorem trintss 4548
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 3109 . . . 4  |-  y  e. 
_V
21elint2 4278 . . 3  |-  ( y  e.  |^| A  <->  A. x  e.  A  y  e.  x )
3 r19.2z 3906 . . . . 5  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  y  e.  x )  ->  E. x  e.  A  y  e.  x )
43ex 432 . . . 4  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  y  e.  x  ->  E. x  e.  A  y  e.  x ) )
5 trel 4539 . . . . . 6  |-  ( Tr  A  ->  ( (
y  e.  x  /\  x  e.  A )  ->  y  e.  A ) )
65expcomd 436 . . . . 5  |-  ( Tr  A  ->  ( x  e.  A  ->  ( y  e.  x  ->  y  e.  A ) ) )
76rexlimdv 2944 . . . 4  |-  ( Tr  A  ->  ( E. x  e.  A  y  e.  x  ->  y  e.  A ) )
84, 7sylan9 655 . . 3  |-  ( ( A  =/=  (/)  /\  Tr  A )  ->  ( A. x  e.  A  y  e.  x  ->  y  e.  A ) )
92, 8syl5bi 217 . 2  |-  ( ( A  =/=  (/)  /\  Tr  A )  ->  (
y  e.  |^| A  ->  y  e.  A ) )
109ssrdv 3495 1  |-  ( ( A  =/=  (/)  /\  Tr  A )  ->  |^| A  C_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805    C_ wss 3461   (/)c0 3783   |^|cint 4271   Tr wtr 4532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-v 3108  df-dif 3464  df-in 3468  df-ss 3475  df-nul 3784  df-uni 4236  df-int 4272  df-tr 4533
This theorem is referenced by: (None)
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