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

Theorem wetrep 4320
Description: An epsilon well-ordering is a transitive relation. (Contributed by NM, 22-Apr-1994.)
Assertion
Ref Expression
wetrep  |-  ( (  _E  We  A  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A
) )  ->  (
( x  e.  y  /\  y  e.  z )  ->  x  e.  z ) )
Distinct variable group:    x, A, y, z

Proof of Theorem wetrep
StepHypRef Expression
1 df-3an 965 . . 3  |-  ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  <->  ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A ) )
2 df-wetr 4294 . . . . . . . . 9  |-  (  _E  We  A  <->  (  _E  Fr  A  /\  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( (
x  _E  y  /\  y  _E  z )  ->  x  _E  z ) ) )
32simprbi 273 . . . . . . . 8  |-  (  _E  We  A  ->  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( (
x  _E  y  /\  y  _E  z )  ->  x  _E  z ) )
43r19.21bi 2545 . . . . . . 7  |-  ( (  _E  We  A  /\  x  e.  A )  ->  A. y  e.  A  A. z  e.  A  ( ( x  _E  y  /\  y  _E  z )  ->  x  _E  z ) )
54r19.21bi 2545 . . . . . 6  |-  ( ( (  _E  We  A  /\  x  e.  A
)  /\  y  e.  A )  ->  A. z  e.  A  ( (
x  _E  y  /\  y  _E  z )  ->  x  _E  z ) )
65anasss 397 . . . . 5  |-  ( (  _E  We  A  /\  ( x  e.  A  /\  y  e.  A
) )  ->  A. z  e.  A  ( (
x  _E  y  /\  y  _E  z )  ->  x  _E  z ) )
76r19.21bi 2545 . . . 4  |-  ( ( (  _E  We  A  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  (
( x  _E  y  /\  y  _E  z
)  ->  x  _E  z ) )
87anasss 397 . . 3  |-  ( (  _E  We  A  /\  ( ( x  e.  A  /\  y  e.  A )  /\  z  e.  A ) )  -> 
( ( x  _E  y  /\  y  _E  z )  ->  x  _E  z ) )
91, 8sylan2b 285 . 2  |-  ( (  _E  We  A  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A
) )  ->  (
( x  _E  y  /\  y  _E  z
)  ->  x  _E  z ) )
10 epel 4252 . . 3  |-  ( x  _E  y  <->  x  e.  y )
11 epel 4252 . . 3  |-  ( y  _E  z  <->  y  e.  z )
1210, 11anbi12i 456 . 2  |-  ( ( x  _E  y  /\  y  _E  z )  <->  ( x  e.  y  /\  y  e.  z )
)
13 epel 4252 . 2  |-  ( x  _E  z  <->  x  e.  z )
149, 12, 133imtr3g 203 1  |-  ( (  _E  We  A  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A
) )  ->  (
( x  e.  y  /\  y  e.  z )  ->  x  e.  z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 963    e. wcel 2128   A.wral 2435   class class class wbr 3965    _E cep 4247    Fr wfr 4288    We wwe 4290
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-eprel 4249  df-wetr 4294
This theorem is referenced by:  wessep  4537
  Copyright terms: Public domain W3C validator