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Theorem trintssm 4101
Description: Any inhabited transitive class includes its intersection. Similar to Exercise 3 in [TakeutiZaring] p. 44 (which mistakenly does not include the inhabitedness hypothesis). (Contributed by Jim Kingdon, 22-Aug-2018.)
Assertion
Ref Expression
trintssm  |-  ( ( Tr  A  /\  E. x  x  e.  A
)  ->  |^| A  C_  A )
Distinct variable group:    x, A

Proof of Theorem trintssm
StepHypRef Expression
1 intss1 3844 . . . 4  |-  ( x  e.  A  ->  |^| A  C_  x )
2 trss 4094 . . . . 5  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
32com12 30 . . . 4  |-  ( x  e.  A  ->  ( Tr  A  ->  x  C_  A ) )
4 sstr2 3154 . . . 4  |-  ( |^| A  C_  x  ->  (
x  C_  A  ->  |^| A  C_  A )
)
51, 3, 4sylsyld 58 . . 3  |-  ( x  e.  A  ->  ( Tr  A  ->  |^| A  C_  A ) )
65exlimiv 1591 . 2  |-  ( E. x  x  e.  A  ->  ( Tr  A  ->  |^| A  C_  A )
)
76impcom 124 1  |-  ( ( Tr  A  /\  E. x  x  e.  A
)  ->  |^| A  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   E.wex 1485    e. wcel 2141    C_ wss 3121   |^|cint 3829   Tr wtr 4085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-in 3127  df-ss 3134  df-uni 3795  df-int 3830  df-tr 4086
This theorem is referenced by: (None)
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