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Theorem wess 4273
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 4253 . . 3  |-  ( A 
C_  B  ->  ( R  Fr  B  ->  R  Fr  A ) )
2 soss 4225 . . 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 4247 . 2  |-  ( R  We  B  <->  ( R  Fr  B  /\  R  Or  B ) )
5 df-we 4247 . 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 3078    Or wor 4206    Fr wfr 4242    We wwe 4244
This theorem is referenced by:  wefrc  4280  trssord  4302  ordelord  4307  fnwelem  6082  ordtypelem8  7124  oismo  7139  cantnfcl  7252  infxpenlem  7525  ac10ct  7545  dfac12lem2  7654  cflim2  7773  cofsmo  7779  hsmexlem1  7936  smobeth  8088  canthwelem  8152  gruina  8320  ltwefz  10904  omsinds  23387  wfrlem5  23428  tfrALTlem  23444  welb  25583  dnwech  26311  aomclem4  26320  dfac11  26326  onfrALTlem3  27002  onfrALTlem3VD  27353
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 1926  ax-ext 2234
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 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ral 2513  df-in 3085  df-ss 3089  df-po 4207  df-so 4208  df-fr 4245  df-we 4247
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