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

Theorem onfrALTlem1 28708
Description: Lemma for onfrALT 28709. (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 1763 . . . . 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 28707 . . . 4  |-  ( E. x ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  <->  E. y [ y  /  x ] ( x  e.  a  /\  ( a  i^i  x
)  =  (/) ) )
42, 3syl6ib 219 . . 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 3167 . . . . 5  |-  ( [ y  /  x ]
( x  e.  a  /\  ( a  i^i  x )  =  (/) ) 
<-> 
[. y  /  x ]. ( x  e.  a  /\  ( a  i^i  x )  =  (/) ) )
6 onfrALTlem4 28703 . . . . 5  |-  ( [. y  /  x ]. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  <->  ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
75, 6bitri 242 . . . 4  |-  ( [ y  /  x ]
( x  e.  a  /\  ( a  i^i  x )  =  (/) ) 
<->  ( y  e.  a  /\  ( a  i^i  y )  =  (/) ) )
87exbii 1593 . . 3  |-  ( E. y [ y  /  x ] ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  <->  E. y ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
94, 8syl6ib 219 . 2  |-  ( ( a  C_  On  /\  a  =/=  (/) )  ->  (
( x  e.  a  /\  ( a  i^i  x )  =  (/) )  ->  E. y ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) ) )
10 df-rex 2713 . 2  |-  ( E. y  e.  a  ( a  i^i  y )  =  (/)  <->  E. y ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
119, 10syl6ibr 220 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 360   E.wex 1551    = wceq 1653   [wsb 1659    =/= wne 2601   E.wrex 2708   [.wsbc 3163    i^i cin 3321    C_ wss 3322   (/)c0 3630   Oncon0 4584
This theorem is referenced by:  onfrALT  28709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-in 3329
  Copyright terms: Public domain W3C validator