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Theorem trint0 4132
Description: Any non-empty 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 3466 . . 3  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
2 intss1 3879 . . . . 5  |-  ( x  e.  A  ->  |^| A  C_  x )
3 trss 4124 . . . . . 6  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
43com12 27 . . . . 5  |-  ( x  e.  A  ->  ( Tr  A  ->  x  C_  A ) )
5 sstr2 3188 . . . . 5  |-  ( |^| A  C_  x  ->  (
x  C_  A  ->  |^| A  C_  A )
)
62, 4, 5sylsyld 52 . . . 4  |-  ( x  e.  A  ->  ( Tr  A  ->  |^| A  C_  A ) )
76exlimiv 1668 . . 3  |-  ( E. x  x  e.  A  ->  ( Tr  A  ->  |^| A  C_  A )
)
81, 7sylbi 187 . 2  |-  ( A  =/=  (/)  ->  ( Tr  A  ->  |^| A  C_  A
) )
98impcom 419 1  |-  ( ( Tr  A  /\  A  =/=  (/) )  ->  |^| A  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1530    e. wcel 1686    =/= wne 2448    C_ wss 3154   (/)c0 3457   |^|cint 3864   Tr wtr 4115
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-v 2792  df-dif 3157  df-in 3161  df-ss 3168  df-nul 3458  df-uni 3830  df-int 3865  df-tr 4116
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