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Theorem trint0 4279
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 3597 . . 3  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
2 intss1 4025 . . . . 5  |-  ( x  e.  A  ->  |^| A  C_  x )
3 trss 4271 . . . . . 6  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
43com12 29 . . . . 5  |-  ( x  e.  A  ->  ( Tr  A  ->  x  C_  A ) )
5 sstr2 3315 . . . . 5  |-  ( |^| A  C_  x  ->  (
x  C_  A  ->  |^| A  C_  A )
)
62, 4, 5sylsyld 54 . . . 4  |-  ( x  e.  A  ->  ( Tr  A  ->  |^| A  C_  A ) )
76exlimiv 1641 . . 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 1547    e. wcel 1721    =/= wne 2567    C_ wss 3280   (/)c0 3588   |^|cint 4010   Tr wtr 4262
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-v 2918  df-dif 3283  df-in 3287  df-ss 3294  df-nul 3589  df-uni 3976  df-int 4011  df-tr 4263
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