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Theorem wess 4352
Description: Subset theorem for the well-ordering predicate. Exercise 4 of [TakeutiZaring] p. 31. (Contributed by NM, 19-Apr-1994.)
Assertion
Ref Expression
wess  |-  ( A 
C_  B  ->  ( R  We  B  ->  R  We  A ) )

Proof of Theorem wess
StepHypRef Expression
1 frss 4332 . . 3  |-  ( A 
C_  B  ->  ( R  Fr  B  ->  R  Fr  A ) )
2 soss 4304 . . 3  |-  ( A 
C_  B  ->  ( R  Or  B  ->  R  Or  A ) )
31, 2anim12d 548 . 2  |-  ( A 
C_  B  ->  (
( R  Fr  B  /\  R  Or  B
)  ->  ( R  Fr  A  /\  R  Or  A ) ) )
4 df-we 4326 . 2  |-  ( R  We  B  <->  ( R  Fr  B  /\  R  Or  B ) )
5 df-we 4326 . 2  |-  ( R  We  A  <->  ( R  Fr  A  /\  R  Or  A ) )
63, 4, 53imtr4g 263 1  |-  ( A 
C_  B  ->  ( R  We  B  ->  R  We  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    C_ wss 3127    Or wor 4285    Fr wfr 4321    We wwe 4323
This theorem is referenced by:  wefrc  4359  trssord  4381  ordelord  4386  fnwelem  6164  ordtypelem8  7208  oismo  7223  cantnfcl  7336  infxpenlem  7609  ac10ct  7629  dfac12lem2  7738  cflim2  7857  cofsmo  7863  hsmexlem1  8020  smobeth  8176  canthwelem  8240  gruina  8408  ltwefz  10992  omsinds  23588  wfrlem5  23629  tfrALTlem  23645  welb  25784  dnwech  26512  aomclem4  26521  dfac11  26527  onfrALTlem3  27362  onfrALTlem3VD  27713
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ral 2523  df-in 3134  df-ss 3141  df-po 4286  df-so 4287  df-fr 4324  df-we 4326
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