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Theorem frss 4359
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 3187 . . . . . 6  |-  ( x 
C_  A  ->  ( A  C_  B  ->  x  C_  B ) )
21com12 27 . . . . 5  |-  ( A 
C_  B  ->  (
x  C_  A  ->  x 
C_  B ) )
32anim1d 547 . . . 4  |-  ( A 
C_  B  ->  (
( x  C_  A  /\  x  =/=  (/) )  -> 
( x  C_  B  /\  x  =/=  (/) ) ) )
43imim1d 69 . . 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 1607 . 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 4351 . 2  |-  ( R  Fr  B  <->  A. x
( ( x  C_  B  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) )
7 df-fr 4351 . 2  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) )
85, 6, 73imtr4g 261 1  |-  ( A 
C_  B  ->  ( R  Fr  B  ->  R  Fr  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1527    =/= wne 2447   A.wral 2544   E.wrex 2545    C_ wss 3153   (/)c0 3456   class class class wbr 4024    Fr wfr 4348
This theorem is referenced by:  freq2  4363  wess  4379  frmin  23646  frrlem5  23689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-clab 2271  df-cleq 2277  df-clel 2280  df-in 3160  df-ss 3167  df-fr 4351
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