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Theorem wessep 4383
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 3017 . . . . . . 7  |-  ( B 
C_  A  ->  (
x  e.  B  ->  x  e.  A )
)
2 ssel 3017 . . . . . . 7  |-  ( B 
C_  A  ->  (
y  e.  B  -> 
y  e.  A ) )
3 ssel 3017 . . . . . . 7  |-  ( B 
C_  A  ->  (
z  e.  B  -> 
z  e.  A ) )
41, 2, 33anim123d 1255 . . . . . 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 4178 . . . . . 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 4110 . . . . . 6  |-  ( x  _E  y  <->  x  e.  y )
10 epel 4110 . . . . . 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 4110 . . . . 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 2456 . 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 4379 . . 3  |-  _E  Fr  B
17 df-wetr 4152 . . 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 886 . 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 924    e. wcel 1438   A.wral 2359    C_ wss 2997   class class class wbr 3837    _E cep 4105    Fr wfr 4146    We wwe 4148
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-eprel 4107  df-frfor 4149  df-frind 4150  df-wetr 4152
This theorem is referenced by: (None)
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