MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  trint0 Unicode version

Theorem trint0 4027
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
StepHypRef Expression
1 n0 3371 . . 3  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
2 intss1 3775 . . . . 5  |-  ( x  e.  A  ->  |^| A  C_  x )
3 trss 4019 . . . . . 6  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
43com12 29 . . . . 5  |-  ( x  e.  A  ->  ( Tr  A  ->  x  C_  A ) )
5 sstr2 3107 . . . . 5  |-  ( |^| A  C_  x  ->  (
x  C_  A  ->  |^| A  C_  A )
)
62, 4, 5sylsyld 54 . . . 4  |-  ( x  e.  A  ->  ( Tr  A  ->  |^| A  C_  A ) )
76exlimiv 2023 . . 3  |-  ( E. x  x  e.  A  ->  ( Tr  A  ->  |^| A  C_  A )
)
81, 7sylbi 189 . 2  |-  ( A  =/=  (/)  ->  ( Tr  A  ->  |^| A  C_  A
) )
98impcom 421 1  |-  ( ( Tr  A  /\  A  =/=  (/) )  ->  |^| A  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   E.wex 1537    e. wcel 1621    =/= wne 2412    C_ wss 3078   (/)c0 3362   |^|cint 3760   Tr wtr 4010
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-v 2729  df-dif 3081  df-in 3085  df-ss 3089  df-nul 3363  df-uni 3728  df-int 3761  df-tr 4011
  Copyright terms: Public domain W3C validator