Theorem frss 5081

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.

Step	Hyp	Ref	Expression
1		sstr2 3610	. . . . . 6 |- ( x C_ A -> ( A C_ B -> x C_ B ) )
2	1	com12 32	. . . . 5 |- ( A C_ B -> ( x C_ A -> x C_ B ) )
3	2	anim1d 588	. . . 4 |- ( A C_ B -> ( ( x C_ A /\ x =/= (/) ) -> ( x C_ B /\ x =/= (/) ) ) )
4	3	imim1d 82	. . 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 ) ) )
5	4	alimdv 1845	. 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 5073	. 2 |- ( R Fr B <-> A. x ( ( x C_ B /\ x =/= (/) ) -> E. y e. x A. z e. x -. z R y ) )
7		df-fr 5073	. 2 |- ( R Fr A <-> A. x ( ( x C_ A /\ x =/= (/) ) -> E. y e. x A. z e. x -. z R y ) )
8	5, 6, 7	3imtr4g 285	1 |- ( A C_ B -> ( R Fr B -> R Fr A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   class class class wbr 4653   Fr wfr 5070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588  df-fr 5073

This theorem is referenced by:  freq2  5085  wess  5101  frmin  31739  frrlem5  31784