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Theorem wepo 4377
Description: A well-ordering is a partial ordering. (Contributed by Jim Kingdon, 23-Sep-2021.)
Assertion
Ref Expression
wepo  |-  ( ( R  We  A  /\  A  e.  V )  ->  R  Po  A )

Proof of Theorem wepo
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wefr 4376 . . . 4  |-  ( R  We  A  ->  R  Fr  A )
2 frirrg 4368 . . . 4  |-  ( ( R  Fr  A  /\  A  e.  V  /\  x  e.  A )  ->  -.  x R x )
31, 2syl3an1 1282 . . 3  |-  ( ( R  We  A  /\  A  e.  V  /\  x  e.  A )  ->  -.  x R x )
433expa 1205 . 2  |-  ( ( ( R  We  A  /\  A  e.  V
)  /\  x  e.  A )  ->  -.  x R x )
5 df-3an 982 . . 3  |-  ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  <->  ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A ) )
6 df-wetr 4352 . . . . . . . . . 10  |-  ( R  We  A  <->  ( R  Fr  A  /\  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( (
x R y  /\  y R z )  ->  x R z ) ) )
76simprbi 275 . . . . . . . . 9  |-  ( R  We  A  ->  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( (
x R y  /\  y R z )  ->  x R z ) )
87adantr 276 . . . . . . . 8  |-  ( ( R  We  A  /\  A  e.  V )  ->  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( ( x R y  /\  y R z )  ->  x R z ) )
98r19.21bi 2578 . . . . . . 7  |-  ( ( ( R  We  A  /\  A  e.  V
)  /\  x  e.  A )  ->  A. y  e.  A  A. z  e.  A  ( (
x R y  /\  y R z )  ->  x R z ) )
109r19.21bi 2578 . . . . . 6  |-  ( ( ( ( R  We  A  /\  A  e.  V
)  /\  x  e.  A )  /\  y  e.  A )  ->  A. z  e.  A  ( (
x R y  /\  y R z )  ->  x R z ) )
1110anasss 399 . . . . 5  |-  ( ( ( R  We  A  /\  A  e.  V
)  /\  ( x  e.  A  /\  y  e.  A ) )  ->  A. z  e.  A  ( ( x R y  /\  y R z )  ->  x R z ) )
1211r19.21bi 2578 . . . 4  |-  ( ( ( ( R  We  A  /\  A  e.  V
)  /\  ( x  e.  A  /\  y  e.  A ) )  /\  z  e.  A )  ->  ( ( x R y  /\  y R z )  ->  x R z ) )
1312anasss 399 . . 3  |-  ( ( ( R  We  A  /\  A  e.  V
)  /\  ( (
x  e.  A  /\  y  e.  A )  /\  z  e.  A
) )  ->  (
( x R y  /\  y R z )  ->  x R
z ) )
145, 13sylan2b 287 . 2  |-  ( ( ( R  We  A  /\  A  e.  V
)  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  -> 
( ( x R y  /\  y R z )  ->  x R z ) )
154, 14ispod 4322 1  |-  ( ( R  We  A  /\  A  e.  V )  ->  R  Po  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    /\ w3a 980    e. wcel 2160   A.wral 2468   class class class wbr 4018    Po wpo 4312    Fr wfr 4346    We wwe 4348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-sep 4136
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-po 4314  df-frfor 4349  df-frind 4350  df-wetr 4352
This theorem is referenced by: (None)
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