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Theorem trint0 4311
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 3629 . . 3  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
2 intss1 4057 . . . . 5  |-  ( x  e.  A  ->  |^| A  C_  x )
3 trss 4303 . . . . . 6  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
43com12 29 . . . . 5  |-  ( x  e.  A  ->  ( Tr  A  ->  x  C_  A ) )
5 sstr2 3347 . . . . 5  |-  ( |^| A  C_  x  ->  (
x  C_  A  ->  |^| A  C_  A )
)
62, 4, 5sylsyld 54 . . . 4  |-  ( x  e.  A  ->  ( Tr  A  ->  |^| A  C_  A ) )
76exlimiv 1644 . . 3  |-  ( E. x  x  e.  A  ->  ( Tr  A  ->  |^| A  C_  A )
)
81, 7sylbi 188 . 2  |-  ( A  =/=  (/)  ->  ( Tr  A  ->  |^| A  C_  A
) )
98impcom 420 1  |-  ( ( Tr  A  /\  A  =/=  (/) )  ->  |^| A  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1550    e. wcel 1725    =/= wne 2598    C_ wss 3312   (/)c0 3620   |^|cint 4042   Tr wtr 4294
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-nul 3621  df-uni 4008  df-int 4043  df-tr 4295
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