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Theorem onfrALTVD 34111
Description: Virtual deduction proof of onfrALT 33734. 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 33734 is onfrALTVD 34111 without virtual deductions and was automatically derived from onfrALTVD 34111.
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 33764 . . . . . 6  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  ( a  C_  On  /\  a  =/=  (/) ) ).
2 simpr 459 . . . . . 6  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  a  =/=  (/) )
31, 2e1a 33826 . . . . 5  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  a  =/=  (/)
).
4 exmid 413 . . . . . . . . . 10  |-  ( ( a  i^i  x )  =  (/)  \/  -.  ( a  i^i  x
)  =  (/) )
5 onfrALTlem1VD 34110 . . . . . . . . . . 11  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  ->.  E. y  e.  a  ( a  i^i  y )  =  (/) ).
65in2an 33807 . . . . . . . . . 10  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. x  e.  a  ->.  ( ( a  i^i  x )  =  (/)  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) ).
7 onfrALTlem2VD 34109 . . . . . . . . . . 11  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  -.  ( a  i^i  x
)  =  (/) )  ->.  E. y  e.  a  ( a  i^i  y )  =  (/) ).
87in2an 33807 . . . . . . . . . 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 33840 . . . . . . . . 9  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. x  e.  a  ->.  E. y  e.  a  ( a  i^i  y
)  =  (/) ).
1211in2 33804 . . . . . . . 8  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  ( x  e.  a  ->  E. y  e.  a  ( a  i^i  y )  =  (/) ) ).
1312gen11 33815 . . . . . . 7  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  A. x
( x  e.  a  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) ).
14 19.23v 1765 . . . . . . . 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 33826 . . . . . 6  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  ( E. x  x  e.  a  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) ).
17 n0 3793 . . . . . 6  |-  ( a  =/=  (/)  <->  E. x  x  e.  a )
18 imbi1 321 . . . . . . 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 33893 . . . . 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 33887 . . . 4  |-  (. (
a  C_  On  /\  a  =/=  (/) )  ->.  E. y  e.  a  ( a  i^i  y )  =  (/) ).
2322in1 33761 . . 3  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  E. y  e.  a  ( a  i^i  y )  =  (/) )
2423ax-gen 1623 . 2  |-  A. a
( ( a  C_  On  /\  a  =/=  (/) )  ->  E. y  e.  a 
( a  i^i  y
)  =  (/) )
25 dfepfr 4853 . . 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 33982 1  |-  _E  Fr  On
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367   A.wal 1396    = wceq 1398   E.wex 1617    e. wcel 1823    =/= wne 2649   E.wrex 2805    i^i cin 3460    C_ wss 3461   (/)c0 3783    _E cep 4778    Fr wfr 4824   Oncon0 4867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-vd1 33760  df-vd2 33768  df-vd3 33780
This theorem is referenced by: (None)
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