MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frss Structured version   Visualization version   Unicode version

Theorem frss 4806
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 3425 . . . . . 6  |-  ( x 
C_  A  ->  ( A  C_  B  ->  x  C_  B ) )
21com12 31 . . . . 5  |-  ( A 
C_  B  ->  (
x  C_  A  ->  x 
C_  B ) )
32anim1d 574 . . . 4  |-  ( A 
C_  B  ->  (
( x  C_  A  /\  x  =/=  (/) )  -> 
( x  C_  B  /\  x  =/=  (/) ) ) )
43imim1d 77 . . 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 1771 . 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 4798 . 2  |-  ( R  Fr  B  <->  A. x
( ( x  C_  B  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) )
7 df-fr 4798 . 2  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) )
85, 6, 73imtr4g 278 1  |-  ( A 
C_  B  ->  ( R  Fr  B  ->  R  Fr  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376   A.wal 1450    =/= wne 2641   A.wral 2756   E.wrex 2757    C_ wss 3390   (/)c0 3722   class class class wbr 4395    Fr wfr 4795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-in 3397  df-ss 3404  df-fr 4798
This theorem is referenced by:  freq2  4810  wess  4826  frmin  30551  frrlem5  30589
  Copyright terms: Public domain W3C validator