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Theorem trel 3889
Description: In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
trel  |-  ( Tr  A  ->  ( ( B  e.  C  /\  C  e.  A )  ->  B  e.  A ) )

Proof of Theorem trel
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dftr2 3884 . 2  |-  ( Tr  A  <->  A. y A. x
( ( y  e.  x  /\  x  e.  A )  ->  y  e.  A ) )
2 eleq12 2118 . . . . . 6  |-  ( ( y  =  B  /\  x  =  C )  ->  ( y  e.  x  <->  B  e.  C ) )
3 eleq1 2116 . . . . . . 7  |-  ( x  =  C  ->  (
x  e.  A  <->  C  e.  A ) )
43adantl 266 . . . . . 6  |-  ( ( y  =  B  /\  x  =  C )  ->  ( x  e.  A  <->  C  e.  A ) )
52, 4anbi12d 450 . . . . 5  |-  ( ( y  =  B  /\  x  =  C )  ->  ( ( y  e.  x  /\  x  e.  A )  <->  ( B  e.  C  /\  C  e.  A ) ) )
6 eleq1 2116 . . . . . 6  |-  ( y  =  B  ->  (
y  e.  A  <->  B  e.  A ) )
76adantr 265 . . . . 5  |-  ( ( y  =  B  /\  x  =  C )  ->  ( y  e.  A  <->  B  e.  A ) )
85, 7imbi12d 227 . . . 4  |-  ( ( y  =  B  /\  x  =  C )  ->  ( ( ( y  e.  x  /\  x  e.  A )  ->  y  e.  A )  <->  ( ( B  e.  C  /\  C  e.  A )  ->  B  e.  A ) ) )
98spc2gv 2660 . . 3  |-  ( ( B  e.  C  /\  C  e.  A )  ->  ( A. y A. x ( ( y  e.  x  /\  x  e.  A )  ->  y  e.  A )  ->  (
( B  e.  C  /\  C  e.  A
)  ->  B  e.  A ) ) )
109pm2.43b 50 . 2  |-  ( A. y A. x ( ( y  e.  x  /\  x  e.  A )  ->  y  e.  A )  ->  ( ( B  e.  C  /\  C  e.  A )  ->  B  e.  A ) )
111, 10sylbi 118 1  |-  ( Tr  A  ->  ( ( B  e.  C  /\  C  e.  A )  ->  B  e.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102   A.wal 1257    = wceq 1259    e. wcel 1409   Tr wtr 3882
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952  df-ss 2959  df-uni 3609  df-tr 3883
This theorem is referenced by:  trel3  3890  trintssmOLD  3899  ordtr1  4153  suctr  4186  trsuc  4187  ordn2lp  4297
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