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Theorem onfrALTVD 33399
Description: Virtual deduction proof of onfrALT 33029. 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 33029 is onfrALTVD 33399 without virtual deductions and was automatically derived from onfrALTVD 33399.
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.)
Assertion
Ref Expression
onfrALTVD  |-  _E  Fr  On

Proof of Theorem onfrALTVD
Dummy variables  x  a  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idn1 33059 . . . . . 6  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  ( a  C_  On  /\  a  =/=  (/) ) ).
2 simpr 461 . . . . . 6  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  a  =/=  (/) )
31, 2e1a 33121 . . . . 5  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  a  =/=  (/)
).
4 exmid 415 . . . . . . . . . 10  |-  ( ( a  i^i  x )  =  (/)  \/  -.  ( a  i^i  x
)  =  (/) )
5 onfrALTlem1VD 33398 . . . . . . . . . . 11  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  ->.  E. y  e.  a  ( a  i^i  y )  =  (/) ).
65in2an 33102 . . . . . . . . . 10  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. x  e.  a  ->.  ( ( a  i^i  x )  =  (/)  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) ).
7 onfrALTlem2VD 33397 . . . . . . . . . . 11  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  E. y  e.  a  ( a  i^i  y )  =  (/) ).
87in2an 33102 . . . . . . . . . 10  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. x  e.  a  ->.  ( -.  (
a  i^i  x )  =  (/)  ->  E. y  e.  a  ( a  i^i  y )  =  (/) ) ).
9 pm2.61 171 . . . . . . . . . . 11  |-  ( ( ( 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
)  =  (/) ) )
109a1i 11 . . . . . . . . . 10  |-  ( ( ( 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
)  =  (/) ) ) )
114, 6, 8, 10e022 33135 . . . . . . . . 9  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. x  e.  a  ->.  E. y  e.  a  ( a  i^i  y
)  =  (/) ).
1211in2 33099 . . . . . . . 8  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  ( x  e.  a  ->  E. y  e.  a  ( a  i^i  y )  =  (/) ) ).
1312gen11 33110 . . . . . . 7  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  A. x
( x  e.  a  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) ).
14 19.23v 1745 . . . . . . . 8  |-  ( 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 )  =  (/) ) )
1514biimpi 194 . . . . . . 7  |-  ( 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 )  =  (/) ) )
1613, 15e1a 33121 . . . . . 6  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  ( E. x  x  e.  a  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) ).
17 n0 3776 . . . . . 6  |-  ( a  =/=  (/)  <->  E. x  x  e.  a )
18 imbi1 323 . . . . . . 7  |-  ( ( 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
)  =  (/) ) ) )
1918biimprcd 225 . . . . . 6  |-  ( ( 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 )  =  (/) ) ) )
2016, 17, 19e10 33188 . . . . 5  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  ( a  =/=  (/)  ->  E. y  e.  a  ( a  i^i  y )  =  (/) ) ).
21 pm2.27 39 . . . . 5  |-  ( a  =/=  (/)  ->  ( (
a  =/=  (/)  ->  E. y  e.  a  ( a  i^i  y )  =  (/) )  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) )
223, 20, 21e11 33182 . . . 4  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  E. y  e.  a  ( a  i^i  y )  =  (/) ).
2322in1 33056 . . 3  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  E. y  e.  a  ( a  i^i  y )  =  (/) )
2423ax-gen 1603 . 2  |-  A. a
( ( a  C_  On  /\  a  =/=  (/) )  ->  E. y  e.  a 
( a  i^i  y
)  =  (/) )
25 dfepfr 4850 . . 3  |-  (  _E  Fr  On  <->  A. a
( ( a  C_  On  /\  a  =/=  (/) )  ->  E. y  e.  a 
( a  i^i  y
)  =  (/) ) )
2625biimpri 206 . 2  |-  ( A. a ( ( a 
C_  On  /\  a  =/=  (/) )  ->  E. y  e.  a  ( a  i^i  y )  =  (/) )  ->  _E  Fr  On )
2724, 26e0a 33277 1  |-  _E  Fr  On
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369   A.wal 1379    = wceq 1381   E.wex 1597    e. wcel 1802    =/= wne 2636   E.wrex 2792    i^i cin 3457    C_ wss 3458   (/)c0 3767    _E cep 4775    Fr wfr 4821   Oncon0 4864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-tr 4527  df-eprel 4777  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-vd1 33055  df-vd2 33063  df-vd3 33075
This theorem is referenced by: (None)
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