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Theorem wepo 4153
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 4152 . . . 4  |-  ( R  We  A  ->  R  Fr  A )
2 frirrg 4144 . . . 4  |-  ( ( R  Fr  A  /\  A  e.  V  /\  x  e.  A )  ->  -.  x R x )
31, 2syl3an1 1205 . . 3  |-  ( ( R  We  A  /\  A  e.  V  /\  x  e.  A )  ->  -.  x R x )
433expa 1141 . 2  |-  ( ( ( R  We  A  /\  A  e.  V
)  /\  x  e.  A )  ->  -.  x R x )
5 df-3an 924 . . 3  |-  ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  <->  ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A ) )
6 df-wetr 4128 . . . . . . . . . 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 269 . . . . . . . . 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 270 . . . . . . . 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 2457 . . . . . . 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 2457 . . . . . 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 391 . . . . 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 2457 . . . 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 391 . . 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 281 . 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 4098 1  |-  ( ( R  We  A  /\  A  e.  V )  ->  R  Po  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    /\ w3a 922    e. wcel 1436   A.wral 2355   class class class wbr 3814    Po wpo 4088    Fr wfr 4122    We wwe 4124
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-v 2616  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-sn 3431  df-pr 3432  df-op 3434  df-br 3815  df-po 4090  df-frfor 4125  df-frind 4126  df-wetr 4128
This theorem is referenced by: (None)
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