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Theorem wessep 4492
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 3091 . . . . . . 7  |-  ( B 
C_  A  ->  (
x  e.  B  ->  x  e.  A )
)
2 ssel 3091 . . . . . . 7  |-  ( B 
C_  A  ->  (
y  e.  B  -> 
y  e.  A ) )
3 ssel 3091 . . . . . . 7  |-  ( B 
C_  A  ->  (
z  e.  B  -> 
z  e.  A ) )
41, 2, 33anim123d 1297 . . . . . 6  |-  ( B 
C_  A  ->  (
( x  e.  B  /\  y  e.  B  /\  z  e.  B
)  ->  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) ) )
54adantl 275 . . . . 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 441 . . . 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 4282 . . . . . 6  |-  ( (  _E  We  A  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A
) )  ->  (
( x  e.  y  /\  y  e.  z )  ->  x  e.  z ) )
87adantlr 468 . . . . 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 4214 . . . . . 6  |-  ( x  _E  y  <->  x  e.  y )
10 epel 4214 . . . . . 6  |-  ( y  _E  z  <->  y  e.  z )
119, 10anbi12i 455 . . . . 5  |-  ( ( x  _E  y  /\  y  _E  z )  <->  ( x  e.  y  /\  y  e.  z )
)
12 epel 4214 . . . . 5  |-  ( x  _E  z  <->  x  e.  z )
138, 11, 123imtr4g 204 . . . 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 2515 . 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 4488 . . 3  |-  _E  Fr  B
17 df-wetr 4256 . . 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 924 . 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 133 1  |-  ( (  _E  We  A  /\  B  C_  A )  ->  _E  We  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 962    e. wcel 1480   A.wral 2416    C_ wss 3071   class class class wbr 3929    _E cep 4209    Fr wfr 4250    We wwe 4252
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-eprel 4211  df-frfor 4253  df-frind 4254  df-wetr 4256
This theorem is referenced by: (None)
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