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Metamath Proof Explorer

Theorem onfrALTlem3

Description: Lemma for onfrALT . (Contributed by Alan Sare, 22-Jul-2012) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion onfrALTlem3 
|- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) ) )

Proof

Step Hyp Ref Expression
1 ssid 
 |-  ( a i^i x ) C_ ( a i^i x )
2 simpr 
 |-  ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> -. ( a i^i x ) = (/) )
3 2 a1i 
 |-  ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> -. ( a i^i x ) = (/) ) )
4 df-ne 
 |-  ( ( a i^i x ) =/= (/) <-> -. ( a i^i x ) = (/) )
5 3 4 imbitrrdi 
 |-  ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> ( a i^i x ) =/= (/) ) )
6 pm3.2 
 |-  ( ( a i^i x ) C_ ( a i^i x ) -> ( ( a i^i x ) =/= (/) -> ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) ) )
7 1 5 6 mpsylsyld 
 |-  ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) ) )
8 vex 
 |-  x e. _V
9 8 inex2 
 |-  ( a i^i x ) e. _V
10 inss2 
 |-  ( a i^i x ) C_ x
11 simpl 
 |-  ( ( a C_ On /\ a =/= (/) ) -> a C_ On )
12 simpl 
 |-  ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> x e. a )
13 ssel 
 |-  ( a C_ On -> ( x e. a -> x e. On ) )
14 11 12 13 syl2im 
 |-  ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> x e. On ) )
15 eloni 
 |-  ( x e. On -> Ord x )
16 14 15 syl6 
 |-  ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> Ord x ) )
17 ordwe 
 |-  ( Ord x -> _E We x )
18 16 17 syl6 
 |-  ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> _E We x ) )
19 wess 
 |-  ( ( a i^i x ) C_ x -> ( _E We x -> _E We ( a i^i x ) ) )
20 10 18 19 mpsylsyld 
 |-  ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> _E We ( a i^i x ) ) )
21 wefr 
 |-  ( _E We ( a i^i x ) -> _E Fr ( a i^i x ) )
22 20 21 syl6 
 |-  ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> _E Fr ( a i^i x ) ) )
23 dfepfr 
 |-  ( _E Fr ( a i^i x ) <-> A. b ( ( b C_ ( a i^i x ) /\ b =/= (/) ) -> E. y e. b ( b i^i y ) = (/) ) )
24 22 23 imbitrdi 
 |-  ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> A. b ( ( b C_ ( a i^i x ) /\ b =/= (/) ) -> E. y e. b ( b i^i y ) = (/) ) ) )
25 spsbc 
 |-  ( ( a i^i x ) e. _V -> ( A. b ( ( b C_ ( a i^i x ) /\ b =/= (/) ) -> E. y e. b ( b i^i y ) = (/) ) -> [. ( a i^i x ) / b ]. ( ( b C_ ( a i^i x ) /\ b =/= (/) ) -> E. y e. b ( b i^i y ) = (/) ) ) )
26 9 24 25 mpsylsyld 
 |-  ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> [. ( a i^i x ) / b ]. ( ( b C_ ( a i^i x ) /\ b =/= (/) ) -> E. y e. b ( b i^i y ) = (/) ) ) )
27 onfrALTlem5 
 |-  ( [. ( a i^i x ) / b ]. ( ( b C_ ( a i^i x ) /\ b =/= (/) ) -> E. y e. b ( b i^i y ) = (/) ) <-> ( ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) -> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) ) )
28 26 27 imbitrdi 
 |-  ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> ( ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) -> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) ) ) )
29 7 28 mpdd 
 |-  ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ -. ( a i^i x ) = (/) ) -> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) ) )