MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  trintss Structured version   Unicode version

Theorem trintss 4551
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 3111 . . . 4  |-  y  e. 
_V
21elint2 4284 . . 3  |-  ( y  e.  |^| A  <->  A. x  e.  A  y  e.  x )
3 r19.2z 3912 . . . . 5  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  y  e.  x )  ->  E. x  e.  A  y  e.  x )
43ex 434 . . . 4  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  y  e.  x  ->  E. x  e.  A  y  e.  x ) )
5 trel 4542 . . . . . 6  |-  ( Tr  A  ->  ( (
y  e.  x  /\  x  e.  A )  ->  y  e.  A ) )
65expcomd 438 . . . . 5  |-  ( Tr  A  ->  ( x  e.  A  ->  ( y  e.  x  ->  y  e.  A ) ) )
76rexlimdv 2948 . . . 4  |-  ( Tr  A  ->  ( E. x  e.  A  y  e.  x  ->  y  e.  A ) )
84, 7sylan9 657 . . 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 3505 1  |-  ( ( A  =/=  (/)  /\  Tr  A )  ->  |^| A  C_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1762    =/= wne 2657   A.wral 2809   E.wrex 2810    C_ wss 3471   (/)c0 3780   |^|cint 4277   Tr wtr 4535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-v 3110  df-dif 3474  df-in 3478  df-ss 3485  df-nul 3781  df-uni 4241  df-int 4278  df-tr 4536
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator