Metamath Proof Explorer

Theorem onfrALTVD

Description: Virtual deduction proof of onfrALT . The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. onfrALT is onfrALTVD without virtual deductions and was automatically derived from onfrALTVD .

1:: |- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. E. y e. a ( a i^i y ) = (/) ).
2:: |- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ ( a i^i x ) = (/) ) ->. E. y e. a ( a i^i y ) = (/) ).
3:1: |- (. ( a C_ On /\ a =/= (/) ) ,. x e. a ->. ( -. ( a i^i x ) = (/) -> E. y e. a ( a i^i y ) = (/) ) ).
4:2: |- (. ( a C_ On /\ a =/= (/) ) ,. x e. a ->. ( ( a i^i x ) = (/) -> E. y e. a ( a i^i y ) = (/) ) ).
5:: |- ( ( a i^i x ) = (/) \/ -. ( a i^i x ) = (/) )
6:5,4,3: |- (. ( a C_ On /\ a =/= (/) ) ,. x e. a ->. E. y e. a ( a i^i y ) = (/) ).
7:6: |- (. ( a C_ On /\ a =/= (/) ) ->. ( x e. a -> E. y e. a ( a i^i y ) = (/) ) ).
8:7: |- (. ( a C_ On /\ a =/= (/) ) ->. A. x ( x e. a -> E. y e. a ( a i^i y ) = (/) ) ).
9:8: |- (. ( a C_ On /\ a =/= (/) ) ->. ( E. x x e. a -> E. y e. a ( a i^i y ) = (/) ) ).
10:: |- ( a =/= (/) <-> E. x x e. a )
11:9,10: |- (. ( a C_ On /\ a =/= (/) ) ->. ( a =/= (/) -> E. y e. a ( a i^i y ) = (/) ) ).
12:: |- (. ( a C_ On /\ a =/= (/) ) ->. ( a C_ On /\ a =/= (/) ) ).
13:12: |- (. ( a C_ On /\ a =/= (/) ) ->. a =/= (/) ).
14:13,11: |- (. ( a C_ On /\ a =/= (/) ) ->. E. y e. a ( a i^i y ) = (/) ).
15:14: |- ( ( a C_ On /\ a =/= (/) ) -> E. y e. a ( a i^i y ) = (/) )
16:15: |- A. a ( ( a C_ On /\ a =/= (/) ) -> E. y e. a ( a i^i y ) = (/) )
qed:16: |- _E Fr On
(Contributed by Alan Sare, 22-Jul-2012) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion onfrALTVD 
|- _E Fr On

Proof

Step Hyp Ref Expression
1 idn1 
 |-  (. ( a C_ On /\ a =/= (/) ) ->. ( a C_ On /\ a =/= (/) ) ).
2 simpr 
 |-  ( ( a C_ On /\ a =/= (/) ) -> a =/= (/) )
3 1 2 e1a 
 |-  (. ( a C_ On /\ a =/= (/) ) ->. a =/= (/) ).
4 exmid 
 |-  ( ( a i^i x ) = (/) \/ -. ( a i^i x ) = (/) )
5 onfrALTlem1VD 
 |-  (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ ( a i^i x ) = (/) ) ->. E. y e. a ( a i^i y ) = (/) ).
6 5 in2an 
 |-  (. ( a C_ On /\ a =/= (/) ) ,. x e. a ->. ( ( a i^i x ) = (/) -> E. y e. a ( a i^i y ) = (/) ) ).
7 onfrALTlem2VD 
 |-  (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ -. ( a i^i x ) = (/) ) ->. E. y e. a ( a i^i y ) = (/) ).
8 7 in2an 
 |-  (. ( a C_ On /\ a =/= (/) ) ,. x e. a ->. ( -. ( a i^i x ) = (/) -> E. y e. a ( a i^i y ) = (/) ) ).
9 pm2.61 
 |-  ( ( ( a i^i x ) = (/) -> E. y e. a ( a i^i y ) = (/) ) -> ( ( -. ( a i^i x ) = (/) -> E. y e. a ( a i^i y ) = (/) ) -> E. y e. a ( a i^i y ) = (/) ) )
10 9 a1i 
 |-  ( ( ( a i^i x ) = (/) \/ -. ( a i^i x ) = (/) ) -> ( ( ( a i^i x ) = (/) -> E. y e. a ( a i^i y ) = (/) ) -> ( ( -. ( a i^i x ) = (/) -> E. y e. a ( a i^i y ) = (/) ) -> E. y e. a ( a i^i y ) = (/) ) ) )
11 4 6 8 10 e022 
 |-  (. ( a C_ On /\ a =/= (/) ) ,. x e. a ->. E. y e. a ( a i^i y ) = (/) ).
12 11 in2 
 |-  (. ( a C_ On /\ a =/= (/) ) ->. ( x e. a -> E. y e. a ( a i^i y ) = (/) ) ).
13 12 gen11 
 |-  (. ( a C_ On /\ a =/= (/) ) ->. A. x ( x e. a -> E. y e. a ( a i^i y ) = (/) ) ).
14 19.23v 
 |-  ( A. x ( x e. a -> E. y e. a ( a i^i y ) = (/) ) <-> ( E. x x e. a -> E. y e. a ( a i^i y ) = (/) ) )
15 14 biimpi 
 |-  ( A. x ( x e. a -> E. y e. a ( a i^i y ) = (/) ) -> ( E. x x e. a -> E. y e. a ( a i^i y ) = (/) ) )
16 13 15 e1a 
 |-  (. ( a C_ On /\ a =/= (/) ) ->. ( E. x x e. a -> E. y e. a ( a i^i y ) = (/) ) ).
17 n0 
 |-  ( a =/= (/) <-> E. x x e. a )
18 imbi1 
 |-  ( ( a =/= (/) <-> E. x x e. a ) -> ( ( a =/= (/) -> E. y e. a ( a i^i y ) = (/) ) <-> ( E. x x e. a -> E. y e. a ( a i^i y ) = (/) ) ) )
19 18 biimprcd 
 |-  ( ( E. x x e. a -> E. y e. a ( a i^i y ) = (/) ) -> ( ( a =/= (/) <-> E. x x e. a ) -> ( a =/= (/) -> E. y e. a ( a i^i y ) = (/) ) ) )
20 16 17 19 e10 
 |-  (. ( a C_ On /\ a =/= (/) ) ->. ( a =/= (/) -> E. y e. a ( a i^i y ) = (/) ) ).
21 pm2.27 
 |-  ( a =/= (/) -> ( ( a =/= (/) -> E. y e. a ( a i^i y ) = (/) ) -> E. y e. a ( a i^i y ) = (/) ) )
22 3 20 21 e11 
 |-  (. ( a C_ On /\ a =/= (/) ) ->. E. y e. a ( a i^i y ) = (/) ).
23 22 in1 
 |-  ( ( a C_ On /\ a =/= (/) ) -> E. y e. a ( a i^i y ) = (/) )
24 23 ax-gen 
 |-  A. a ( ( a C_ On /\ a =/= (/) ) -> E. y e. a ( a i^i y ) = (/) )
25 dfepfr 
 |-  ( _E Fr On <-> A. a ( ( a C_ On /\ a =/= (/) ) -> E. y e. a ( a i^i y ) = (/) ) )
26 25 biimpri 
 |-  ( A. a ( ( a C_ On /\ a =/= (/) ) -> E. y e. a ( a i^i y ) = (/) ) -> _E Fr On )
27 24 26 e0a 
 |-  _E Fr On