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

Theorem onfrALTlem1 28313
Description: Lemma for onfrALT 28314. (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 1718 . . . . 5  |-  ( ( x  e.  a  /\  ( a  i^i  x
)  =  (/) )  ->  E. x ( x  e.  a  /\  ( a  i^i  x )  =  (/) ) )
21a1i 10 . . . 4  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  ( a  i^i  x )  =  (/) )  ->  E. x ( x  e.  a  /\  (
a  i^i  x )  =  (/) ) ) )
3 cbvexsv 28312 . . . 4  |-  ( E. x ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  <->  E. y [ y  /  x ] ( x  e.  a  /\  ( a  i^i  x
)  =  (/) ) )
42, 3syl6ib 217 . . 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 2995 . . . . 5  |-  ( [ y  /  x ]
( x  e.  a  /\  ( a  i^i  x )  =  (/) ) 
<-> 
[. y  /  x ]. ( x  e.  a  /\  ( a  i^i  x )  =  (/) ) )
6 onfrALTlem4 28308 . . . . 5  |-  ( [. y  /  x ]. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  <->  ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
75, 6bitri 240 . . . 4  |-  ( [ y  /  x ]
( x  e.  a  /\  ( a  i^i  x )  =  (/) ) 
<->  ( y  e.  a  /\  ( a  i^i  y )  =  (/) ) )
87exbii 1569 . . 3  |-  ( E. y [ y  /  x ] ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  <->  E. y ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
94, 8syl6ib 217 . 2  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  ( a  i^i  x )  =  (/) )  ->  E. y ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) ) )
10 df-rex 2549 . 2  |-  ( E. y  e.  a  ( a  i^i  y )  =  (/)  <->  E. y ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
119, 10syl6ibr 218 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 358   E.wex 1528    = wceq 1623   [wsb 1629    =/= wne 2446   E.wrex 2544   [.wsbc 2991    i^i cin 3151    C_ wss 3152   (/)c0 3455   Oncon0 4392
This theorem is referenced by:  onfrALT  28314
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-in 3159
  Copyright terms: Public domain W3C validator