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Theorem trint0 4507
Description: Any nonempty transitive class includes its intersection. Exercise 2 in [TakeutiZaring] p. 44. (Contributed by Andrew Salmon, 14-Nov-2011.)
Assertion
Ref Expression
trint0  |-  ( ( Tr  A  /\  A  =/=  (/) )  ->  |^| A  C_  A )

Proof of Theorem trint0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 n0 3732 . . 3  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
2 intss1 4241 . . . . 5  |-  ( x  e.  A  ->  |^| A  C_  x )
3 trss 4499 . . . . . 6  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
43com12 31 . . . . 5  |-  ( x  e.  A  ->  ( Tr  A  ->  x  C_  A ) )
5 sstr2 3425 . . . . 5  |-  ( |^| A  C_  x  ->  (
x  C_  A  ->  |^| A  C_  A )
)
62, 4, 5sylsyld 57 . . . 4  |-  ( x  e.  A  ->  ( Tr  A  ->  |^| A  C_  A ) )
76exlimiv 1784 . . 3  |-  ( E. x  x  e.  A  ->  ( Tr  A  ->  |^| A  C_  A )
)
81, 7sylbi 200 . 2  |-  ( A  =/=  (/)  ->  ( Tr  A  ->  |^| A  C_  A
) )
98impcom 437 1  |-  ( ( Tr  A  /\  A  =/=  (/) )  ->  |^| A  C_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376   E.wex 1671    e. wcel 1904    =/= wne 2641    C_ wss 3390   (/)c0 3722   |^|cint 4226   Tr wtr 4490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-v 3033  df-dif 3393  df-in 3397  df-ss 3404  df-nul 3723  df-uni 4191  df-int 4227  df-tr 4491
This theorem is referenced by: (None)
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