Theorem bnj1053

Description: Technical lemma for bnj69 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj1053.1	|- ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
	bnj1053.2	|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
	bnj1053.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
	bnj1053.4	|- ( th <-> ( R _FrSe A /\ X e. A ) )
	bnj1053.5	|- ( ta <-> ( B e. _V /\ _TrFo ( B , A , R ) /\ _pred ( X , A , R ) C_ B ) )
	bnj1053.6	|- ( ze <-> ( i e. n /\ z e. ( f ` i ) ) )
	bnj1053.7	|- D = ( _om \ { (/) } )
	bnj1053.8	|- K = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
	bnj1053.9	|- ( et <-> ( ( th /\ ta /\ ch /\ ze ) -> z e. B ) )
	bnj1053.10	|- ( rh <-> A. j e. n ( j _E i -> [. j / i ]. et ) )
	bnj1053.37	|- ( ( th /\ ta /\ ch /\ ze ) -> A. i e. n ( rh -> et ) )
Assertion	bnj1053	|- ( ( th /\ ta ) -> _trCl ( X , A , R ) C_ B )

Proof

Step	Hyp	Ref	Expression
1		bnj1053.1	|- ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
2		bnj1053.2	|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
3		bnj1053.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
4		bnj1053.4	|- ( th <-> ( R _FrSe A /\ X e. A ) )
5		bnj1053.5	|- ( ta <-> ( B e. _V /\ _TrFo ( B , A , R ) /\ _pred ( X , A , R ) C_ B ) )
6		bnj1053.6	|- ( ze <-> ( i e. n /\ z e. ( f ` i ) ) )
7		bnj1053.7	|- D = ( _om \ { (/) } )
8		bnj1053.8	|- K = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
9		bnj1053.9	|- ( et <-> ( ( th /\ ta /\ ch /\ ze ) -> z e. B ) )
10		bnj1053.10	|- ( rh <-> A. j e. n ( j _E i -> [. j / i ]. et ) )
11		bnj1053.37	|- ( ( th /\ ta /\ ch /\ ze ) -> A. i e. n ( rh -> et ) )
12	7	bnj923	|- ( n e. D -> n e. _om )
13		nnord	|- ( n e. _om -> Ord n )
14		ordfr	|- ( Ord n -> _E Fr n )
15	12 13 14	3syl	|- ( n e. D -> _E Fr n )
16	3 15	bnj769	|- ( ch -> _E Fr n )
17	16	bnj707	|- ( ( th /\ ta /\ ch /\ ze ) -> _E Fr n )
18	17 11	jca	|- ( ( th /\ ta /\ ch /\ ze ) -> ( _E Fr n /\ A. i e. n ( rh -> et ) ) )
19	1 2 3 4 5 6 7 8 9 10 18	bnj1052	|- ( ( th /\ ta ) -> _trCl ( X , A , R ) C_ B )