Theorem bnj1034

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	bnj1034.1	|- ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
	bnj1034.2	|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
	bnj1034.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
	bnj1034.4	|- ( th <-> ( R _FrSe A /\ X e. A ) )
	bnj1034.5	|- ( ta <-> ( B e. _V /\ _TrFo ( B , A , R ) /\ _pred ( X , A , R ) C_ B ) )
	bnj1034.7	|- ( ze <-> ( i e. n /\ z e. ( f ` i ) ) )
	bnj1034.8	|- D = ( _om \ { (/) } )
	bnj1034.9	|- K = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
	bnj1034.10	|- ( E. f E. n E. i ( th /\ ta /\ ch /\ ze ) -> z e. B )
Assertion	bnj1034	|- ( ( th /\ ta ) -> _trCl ( X , A , R ) C_ B )

Proof

Step	Hyp	Ref	Expression
1		bnj1034.1	|- ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
2		bnj1034.2	|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
3		bnj1034.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
4		bnj1034.4	|- ( th <-> ( R _FrSe A /\ X e. A ) )
5		bnj1034.5	|- ( ta <-> ( B e. _V /\ _TrFo ( B , A , R ) /\ _pred ( X , A , R ) C_ B ) )
6		bnj1034.7	|- ( ze <-> ( i e. n /\ z e. ( f ` i ) ) )
7		bnj1034.8	|- D = ( _om \ { (/) } )
8		bnj1034.9	|- K = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
9		bnj1034.10	|- ( E. f E. n E. i ( th /\ ta /\ ch /\ ze ) -> z e. B )
10		biid	|- ( z e. _trCl ( X , A , R ) <-> z e. _trCl ( X , A , R ) )
11	1 2 3 4 5 10 6 7 8 9	bnj1033	|- ( ( th /\ ta ) -> _trCl ( X , A , R ) C_ B )