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Theorem wetrep 4252
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 949 . . 3  |-  ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  <->  ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A ) )
2 df-wetr 4226 . . . . . . . . 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 2497 . . . . . . 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 2497 . . . . . 6  |-  ( ( (  _E  We  A  /\  x  e.  A
)  /\  y  e.  A )  ->  A. z  e.  A  ( (
x  _E  y  /\  y  _E  z )  ->  x  _E  z ) )
65anasss 396 . . . . 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 2497 . . . 4  |-  ( ( (  _E  We  A  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  (
( x  _E  y  /\  y  _E  z
)  ->  x  _E  z ) )
87anasss 396 . . 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 4184 . . 3  |-  ( x  _E  y  <->  x  e.  y )
11 epel 4184 . . 3  |-  ( y  _E  z  <->  y  e.  z )
1210, 11anbi12i 455 . 2  |-  ( ( x  _E  y  /\  y  _E  z )  <->  ( x  e.  y  /\  y  e.  z )
)
13 epel 4184 . 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 947    e. wcel 1465   A.wral 2393   class class class wbr 3899    _E cep 4179    Fr wfr 4220    We wwe 4222
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-eprel 4181  df-wetr 4226
This theorem is referenced by:  wessep  4462
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