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Theorem frss 4297
Description: Subset theorem for the well-founded predicate. Exercise 1 of [TakeutiZaring] p. 31. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
frss  |-  ( A 
C_  B  ->  ( R  Fr  B  ->  R  Fr  A ) )

Proof of Theorem frss
StepHypRef Expression
1 sstr2 3128 . . . . . 6  |-  ( x 
C_  A  ->  ( A  C_  B  ->  x  C_  B ) )
21com12 29 . . . . 5  |-  ( A 
C_  B  ->  (
x  C_  A  ->  x 
C_  B ) )
32anim1d 549 . . . 4  |-  ( A 
C_  B  ->  (
( x  C_  A  /\  x  =/=  (/) )  -> 
( x  C_  B  /\  x  =/=  (/) ) ) )
43imim1d 71 . . 3  |-  ( A 
C_  B  ->  (
( ( x  C_  B  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y )  ->  ( ( x 
C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) ) )
54alimdv 2018 . 2  |-  ( A 
C_  B  ->  ( A. x ( ( x 
C_  B  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y )  ->  A. x ( ( x 
C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) ) )
6 df-fr 4289 . 2  |-  ( R  Fr  B  <->  A. x
( ( x  C_  B  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) )
7 df-fr 4289 . 2  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) )
85, 6, 73imtr4g 263 1  |-  ( A 
C_  B  ->  ( R  Fr  B  ->  R  Fr  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360   A.wal 1532    =/= wne 2419   A.wral 2516   E.wrex 2517    C_ wss 3094   (/)c0 3397   class class class wbr 3963    Fr wfr 4286
This theorem is referenced by:  freq2  4301  wess  4317  frmin  23576  frrlem5  23619
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 2237
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-in 3101  df-ss 3108  df-fr 4289
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