Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  onfrALTlem1 Unicode version

Theorem onfrALTlem1 28349
Description: Lemma for onfrALT 28350. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
onfrALTlem1  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  ( a  i^i  x )  =  (/) )  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) )
Distinct variable group:    x, a, y

Proof of Theorem onfrALTlem1
StepHypRef Expression
1 19.8a 1758 . . . . 5  |-  ( ( x  e.  a  /\  ( a  i^i  x
)  =  (/) )  ->  E. x ( x  e.  a  /\  ( a  i^i  x )  =  (/) ) )
21a1i 11 . . . 4  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  ( a  i^i  x )  =  (/) )  ->  E. x ( x  e.  a  /\  (
a  i^i  x )  =  (/) ) ) )
3 cbvexsv 28348 . . . 4  |-  ( E. x ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  <->  E. y [ y  /  x ] ( x  e.  a  /\  ( a  i^i  x
)  =  (/) ) )
42, 3syl6ib 218 . . 3  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  ( a  i^i  x )  =  (/) )  ->  E. y [ y  /  x ] ( x  e.  a  /\  ( a  i^i  x
)  =  (/) ) ) )
5 sbsbc 3129 . . . . 5  |-  ( [ y  /  x ]
( x  e.  a  /\  ( a  i^i  x )  =  (/) ) 
<-> 
[. y  /  x ]. ( x  e.  a  /\  ( a  i^i  x )  =  (/) ) )
6 onfrALTlem4 28344 . . . . 5  |-  ( [. y  /  x ]. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  <->  ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
75, 6bitri 241 . . . 4  |-  ( [ y  /  x ]
( x  e.  a  /\  ( a  i^i  x )  =  (/) ) 
<->  ( y  e.  a  /\  ( a  i^i  y )  =  (/) ) )
87exbii 1589 . . 3  |-  ( E. y [ y  /  x ] ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  <->  E. y ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
94, 8syl6ib 218 . 2  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  ( a  i^i  x )  =  (/) )  ->  E. y ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) ) )
10 df-rex 2676 . 2  |-  ( E. y  e.  a  ( a  i^i  y )  =  (/)  <->  E. y ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
119, 10syl6ibr 219 1  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  ( a  i^i  x )  =  (/) )  ->  E. y  e.  a  ( a  i^i  y
)  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649   [wsb 1655    =/= wne 2571   E.wrex 2671   [.wsbc 3125    i^i cin 3283    C_ wss 3284   (/)c0 3592   Oncon0 4545
This theorem is referenced by:  onfrALT  28350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-in 3291
  Copyright terms: Public domain W3C validator