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Theorem trintssm 3952
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 3703 . . . 4  |-  ( x  e.  A  ->  |^| A  C_  x )
2 trss 3945 . . . . 5  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
32com12 30 . . . 4  |-  ( x  e.  A  ->  ( Tr  A  ->  x  C_  A ) )
4 sstr2 3032 . . . 4  |-  ( |^| A  C_  x  ->  (
x  C_  A  ->  |^| A  C_  A )
)
51, 3, 4sylsyld 57 . . 3  |-  ( x  e.  A  ->  ( Tr  A  ->  |^| A  C_  A ) )
65exlimiv 1534 . 2  |-  ( E. x  x  e.  A  ->  ( Tr  A  ->  |^| A  C_  A )
)
76impcom 123 1  |-  ( ( Tr  A  /\  E. x  x  e.  A
)  ->  |^| A  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   E.wex 1426    e. wcel 1438    C_ wss 2999   |^|cint 3688   Tr wtr 3936
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-in 3005  df-ss 3012  df-uni 3654  df-int 3689  df-tr 3937
This theorem is referenced by: (None)
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