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Theorem ordwe 4458
Description: Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.)
Assertion
Ref Expression
ordwe  |-  ( Ord 
A  ->  _E  We  A )

Proof of Theorem ordwe
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordfr 4457 . 2  |-  ( Ord 
A  ->  _E  Fr  A )
2 ordelord 4271 . . . . 5  |-  ( ( Ord  A  /\  z  e.  A )  ->  Ord  z )
323ad2antr3 1131 . . . 4  |-  ( ( Ord  A  /\  (
x  e.  A  /\  y  e.  A  /\  z  e.  A )
)  ->  Ord  z )
4 ordtr1 4278 . . . . 5  |-  ( Ord  z  ->  ( (
x  e.  y  /\  y  e.  z )  ->  x  e.  z ) )
5 epel 4182 . . . . . 6  |-  ( x  _E  y  <->  x  e.  y )
6 epel 4182 . . . . . 6  |-  ( y  _E  z  <->  y  e.  z )
75, 6anbi12i 453 . . . . 5  |-  ( ( x  _E  y  /\  y  _E  z )  <->  ( x  e.  y  /\  y  e.  z )
)
8 epel 4182 . . . . 5  |-  ( x  _E  z  <->  x  e.  z )
94, 7, 83imtr4g 204 . . . 4  |-  ( Ord  z  ->  ( (
x  _E  y  /\  y  _E  z )  ->  x  _E  z ) )
103, 9syl 14 . . 3  |-  ( ( Ord  A  /\  (
x  e.  A  /\  y  e.  A  /\  z  e.  A )
)  ->  ( (
x  _E  y  /\  y  _E  z )  ->  x  _E  z ) )
1110ralrimivvva 2490 . 2  |-  ( Ord 
A  ->  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( (
x  _E  y  /\  y  _E  z )  ->  x  _E  z ) )
12 df-wetr 4224 . 2  |-  (  _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 ) ) )
131, 11, 12sylanbrc 411 1  |-  ( Ord 
A  ->  _E  We  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 945    e. wcel 1463   A.wral 2391   class class class wbr 3897    _E cep 4177    Fr wfr 4218    We wwe 4220   Ord word 4252
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-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-tr 3995  df-eprel 4179  df-frfor 4221  df-frind 4222  df-wetr 4224  df-iord 4256
This theorem is referenced by:  nnwetri  6770
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