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Theorem onfrALTVD 28667
Description: Virtual deduction proof of onfrALT 28314. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. onfrALT 28314 is onfrALTVD 28667 without virtual deductions and was automatically derived from onfrALTVD 28667.
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 28342 . . . . . 6  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  ( a  C_  On  /\  a  =/=  (/) ) ).
2 simpr 447 . . . . . 6  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  a  =/=  (/) )
31, 2e1_ 28399 . . . . 5  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  a  =/=  (/)
).
4 exmid 404 . . . . . . . . . 10  |-  ( ( a  i^i  x )  =  (/)  \/  -.  ( a  i^i  x
)  =  (/) )
5 onfrALTlem1VD 28666 . . . . . . . . . . 11  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  ->.  E. y  e.  a  ( a  i^i  y )  =  (/) ).
65in2an 28380 . . . . . . . . . 10  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. x  e.  a  ->.  ( ( a  i^i  x )  =  (/)  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) ).
7 onfrALTlem2VD 28665 . . . . . . . . . . 11  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  E. y  e.  a  ( a  i^i  y )  =  (/) ).
87in2an 28380 . . . . . . . . . 10  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. x  e.  a  ->.  ( -.  (
a  i^i  x )  =  (/)  ->  E. y  e.  a  ( a  i^i  y )  =  (/) ) ).
9 pm2.61 163 . . . . . . . . . . 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 10 . . . . . . . . . 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 28413 . . . . . . . . 9  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. x  e.  a  ->.  E. y  e.  a  ( a  i^i  y
)  =  (/) ).
1211in2 28377 . . . . . . . 8  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  ( x  e.  a  ->  E. y  e.  a  ( a  i^i  y )  =  (/) ) ).
1312gen11 28388 . . . . . . 7  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  A. x
( x  e.  a  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) ).
14 19.23v 1832 . . . . . . . 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 186 . . . . . . 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, 15e1_ 28399 . . . . . 6  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  ( E. x  x  e.  a  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) ).
17 n0 3464 . . . . . 6  |-  ( a  =/=  (/)  <->  E. x  x  e.  a )
18 imbi1 313 . . . . . . 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 216 . . . . . 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 28467 . . . . 5  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  ( a  =/=  (/)  ->  E. y  e.  a  ( a  i^i  y )  =  (/) ) ).
21 pm2.27 35 . . . . 5  |-  ( a  =/=  (/)  ->  ( (
a  =/=  (/)  ->  E. y  e.  a  ( a  i^i  y )  =  (/) )  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) )
223, 20, 21e11 28460 . . . 4  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  E. y  e.  a  ( a  i^i  y )  =  (/) ).
2322in1 28339 . . 3  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  E. y  e.  a  ( a  i^i  y )  =  (/) )
2423ax-gen 1533 . 2  |-  A. a
( ( a  C_  On  /\  a  =/=  (/) )  ->  E. y  e.  a 
( a  i^i  y
)  =  (/) )
25 dfepfr 4378 . . 3  |-  (  _E  Fr  On  <->  A. a
( ( a  C_  On  /\  a  =/=  (/) )  ->  E. y  e.  a 
( a  i^i  y
)  =  (/) ) )
2625biimpri 197 . 2  |-  ( A. a ( ( a 
C_  On  /\  a  =/=  (/) )  ->  E. y  e.  a  ( a  i^i  y )  =  (/) )  ->  _E  Fr  On )
2724, 26e0_ 28547 1  |-  _E  Fr  On
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    i^i cin 3151    C_ wss 3152   (/)c0 3455    _E cep 4303    Fr wfr 4349   Oncon0 4392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-vd1 28338  df-vd2 28347  df-vd3 28359
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