ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  trel Unicode version

Theorem trel 4001
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 3996 . 2  |-  ( Tr  A  <->  A. y A. x
( ( y  e.  x  /\  x  e.  A )  ->  y  e.  A ) )
2 eleq12 2180 . . . . . 6  |-  ( ( y  =  B  /\  x  =  C )  ->  ( y  e.  x  <->  B  e.  C ) )
3 eleq1 2178 . . . . . . 7  |-  ( x  =  C  ->  (
x  e.  A  <->  C  e.  A ) )
43adantl 273 . . . . . 6  |-  ( ( y  =  B  /\  x  =  C )  ->  ( x  e.  A  <->  C  e.  A ) )
52, 4anbi12d 462 . . . . 5  |-  ( ( y  =  B  /\  x  =  C )  ->  ( ( y  e.  x  /\  x  e.  A )  <->  ( B  e.  C  /\  C  e.  A ) ) )
6 eleq1 2178 . . . . . 6  |-  ( y  =  B  ->  (
y  e.  A  <->  B  e.  A ) )
76adantr 272 . . . . 5  |-  ( ( y  =  B  /\  x  =  C )  ->  ( y  e.  A  <->  B  e.  A ) )
85, 7imbi12d 233 . . . 4  |-  ( ( y  =  B  /\  x  =  C )  ->  ( ( ( y  e.  x  /\  x  e.  A )  ->  y  e.  A )  <->  ( ( B  e.  C  /\  C  e.  A )  ->  B  e.  A ) ) )
98spc2gv 2748 . . 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 52 . 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 120 1  |-  ( Tr  A  ->  ( ( B  e.  C  /\  C  e.  A )  ->  B  e.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1312    = wceq 1314    e. wcel 1463   Tr wtr 3994
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-in 3045  df-ss 3052  df-uni 3705  df-tr 3995
This theorem is referenced by:  trel3  4002  ordtr1  4278  suctr  4311  trsuc  4312  ordn2lp  4428
  Copyright terms: Public domain W3C validator