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Theorem wessep 4349
Description: A subset of a set well-ordered by set membership is well-ordered by set membership. (Contributed by Jim Kingdon, 30-Sep-2021.)
Assertion
Ref Expression
wessep  |-  ( (  _E  We  A  /\  B  C_  A )  ->  _E  We  B )

Proof of Theorem wessep
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3003 . . . . . . 7  |-  ( B 
C_  A  ->  (
x  e.  B  ->  x  e.  A )
)
2 ssel 3003 . . . . . . 7  |-  ( B 
C_  A  ->  (
y  e.  B  -> 
y  e.  A ) )
3 ssel 3003 . . . . . . 7  |-  ( B 
C_  A  ->  (
z  e.  B  -> 
z  e.  A ) )
41, 2, 33anim123d 1251 . . . . . 6  |-  ( B 
C_  A  ->  (
( x  e.  B  /\  y  e.  B  /\  z  e.  B
)  ->  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) ) )
54adantl 271 . . . . 5  |-  ( (  _E  We  A  /\  B  C_  A )  -> 
( ( x  e.  B  /\  y  e.  B  /\  z  e.  B )  ->  (
x  e.  A  /\  y  e.  A  /\  z  e.  A )
) )
65imdistani 434 . . . 4  |-  ( ( (  _E  We  A  /\  B  C_  A )  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( (  _E  We  A  /\  B  C_  A
)  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) ) )
7 wetrep 4144 . . . . . 6  |-  ( (  _E  We  A  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A
) )  ->  (
( x  e.  y  /\  y  e.  z )  ->  x  e.  z ) )
87adantlr 461 . . . . 5  |-  ( ( (  _E  We  A  /\  B  C_  A )  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  -> 
( ( x  e.  y  /\  y  e.  z )  ->  x  e.  z ) )
9 epel 4076 . . . . . 6  |-  ( x  _E  y  <->  x  e.  y )
10 epel 4076 . . . . . 6  |-  ( y  _E  z  <->  y  e.  z )
119, 10anbi12i 448 . . . . 5  |-  ( ( x  _E  y  /\  y  _E  z )  <->  ( x  e.  y  /\  y  e.  z )
)
12 epel 4076 . . . . 5  |-  ( x  _E  z  <->  x  e.  z )
138, 11, 123imtr4g 203 . . . 4  |-  ( ( (  _E  We  A  /\  B  C_  A )  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  -> 
( ( x  _E  y  /\  y  _E  z )  ->  x  _E  z ) )
146, 13syl 14 . . 3  |-  ( ( (  _E  We  A  /\  B  C_  A )  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  _E  y  /\  y  _E  z )  ->  x  _E  z ) )
1514ralrimivvva 2449 . 2  |-  ( (  _E  We  A  /\  B  C_  A )  ->  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( ( x  _E  y  /\  y  _E  z )  ->  x  _E  z ) )
16 zfregfr 4345 . . 3  |-  _E  Fr  B
17 df-wetr 4118 . . 3  |-  (  _E  We  B  <->  (  _E  Fr  B  /\  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( (
x  _E  y  /\  y  _E  z )  ->  x  _E  z ) ) )
1816, 17mpbiran 882 . 2  |-  (  _E  We  B  <->  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( (
x  _E  y  /\  y  _E  z )  ->  x  _E  z ) )
1915, 18sylibr 132 1  |-  ( (  _E  We  A  /\  B  C_  A )  ->  _E  We  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 920    e. wcel 1434   A.wral 2353    C_ wss 2983   class class class wbr 3806    _E cep 4071    Fr wfr 4112    We wwe 4114
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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-setind 4309
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2612  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-br 3807  df-opab 3861  df-eprel 4073  df-frfor 4115  df-frind 4116  df-wetr 4118
This theorem is referenced by: (None)
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