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Theorem ordwe 4460
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 4459 . 2  |-  ( Ord 
A  ->  _E  Fr  A )
2 ordelord 4273 . . . . 5  |-  ( ( Ord  A  /\  z  e.  A )  ->  Ord  z )
323ad2antr3 1133 . . . 4  |-  ( ( Ord  A  /\  (
x  e.  A  /\  y  e.  A  /\  z  e.  A )
)  ->  Ord  z )
4 ordtr1 4280 . . . . 5  |-  ( Ord  z  ->  ( (
x  e.  y  /\  y  e.  z )  ->  x  e.  z ) )
5 epel 4184 . . . . . 6  |-  ( x  _E  y  <->  x  e.  y )
6 epel 4184 . . . . . 6  |-  ( y  _E  z  <->  y  e.  z )
75, 6anbi12i 455 . . . . 5  |-  ( ( x  _E  y  /\  y  _E  z )  <->  ( x  e.  y  /\  y  e.  z )
)
8 epel 4184 . . . . 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 2492 . 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 4226 . 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 413 1  |-  ( Ord 
A  ->  _E  We  A )
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   Ord word 4254
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  ax-setind 4422
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-rex 2399  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-uni 3707  df-br 3900  df-opab 3960  df-tr 3997  df-eprel 4181  df-frfor 4223  df-frind 4224  df-wetr 4226  df-iord 4258
This theorem is referenced by:  nnwetri  6772
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