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Theorem frss 4674
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
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sstr2 3351 . . . . . 6  |-  ( x 
C_  A  ->  ( A  C_  B  ->  x  C_  B ) )
21com12 31 . . . . 5  |-  ( A 
C_  B  ->  (
x  C_  A  ->  x 
C_  B ) )
32anim1d 559 . . . 4  |-  ( A 
C_  B  ->  (
( x  C_  A  /\  x  =/=  (/) )  -> 
( x  C_  B  /\  x  =/=  (/) ) ) )
43imim1d 75 . . 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 1674 . 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 4666 . 2  |-  ( R  Fr  B  <->  A. x
( ( x  C_  B  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) )
7 df-fr 4666 . 2  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) )
85, 6, 73imtr4g 270 1  |-  ( A 
C_  B  ->  ( R  Fr  B  ->  R  Fr  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369   A.wal 1360    =/= wne 2596   A.wral 2705   E.wrex 2706    C_ wss 3316   (/)c0 3625   class class class wbr 4280    Fr wfr 4663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-in 3323  df-ss 3330  df-fr 4666
This theorem is referenced by:  freq2  4678  wess  4694  frmin  27550  frrlem5  27619
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