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Theorem wepo 4332
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 4331 . . . 4  |-  ( R  We  A  ->  R  Fr  A )
2 frirrg 4323 . . . 4  |-  ( ( R  Fr  A  /\  A  e.  V  /\  x  e.  A )  ->  -.  x R x )
31, 2syl3an1 1260 . . 3  |-  ( ( R  We  A  /\  A  e.  V  /\  x  e.  A )  ->  -.  x R x )
433expa 1192 . 2  |-  ( ( ( R  We  A  /\  A  e.  V
)  /\  x  e.  A )  ->  -.  x R x )
5 df-3an 969 . . 3  |-  ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  <->  ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A ) )
6 df-wetr 4307 . . . . . . . . . 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 273 . . . . . . . . 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 274 . . . . . . . 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 2552 . . . . . . 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 2552 . . . . . 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 397 . . . . 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 2552 . . . 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 397 . . 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 285 . 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 4277 1  |-  ( ( R  We  A  /\  A  e.  V )  ->  R  Po  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    /\ w3a 967    e. wcel 2135   A.wral 2442   class class class wbr 3977    Po wpo 4267    Fr wfr 4301    We wwe 4303
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-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146  ax-sep 4095
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-v 2724  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125  df-sn 3577  df-pr 3578  df-op 3580  df-br 3978  df-po 4269  df-frfor 4304  df-frind 4305  df-wetr 4307
This theorem is referenced by: (None)
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