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Theorem oi0 7497
Description: Definition of the ordinal isomorphism when its arguments are not meaningful. (Contributed by Mario Carneiro, 25-Jun-2015.)
Hypothesis
Ref Expression
oicl.1  |-  F  = OrdIso
( R ,  A
)
Assertion
Ref Expression
oi0  |-  ( -.  ( R  We  A  /\  R Se  A )  ->  F  =  (/) )

Proof of Theorem oi0
Dummy variables  u  t  v  x  h  j  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oicl.1 . . 3  |-  F  = OrdIso
( R ,  A
)
2 df-oi 7479 . . 3  |- OrdIso ( R ,  A )  =  if ( ( R  We  A  /\  R Se  A ) ,  (recs ( ( h  e. 
_V  |->  ( iota_ v  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v ) ) )  |`  { x  e.  On  |  E. t  e.  A  A. z  e.  (recs ( ( h  e. 
_V  |->  ( iota_ v  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v ) ) )
" x ) z R t } ) ,  (/) )
31, 2eqtri 2456 . 2  |-  F  =  if ( ( R  We  A  /\  R Se  A ) ,  (recs ( ( h  e. 
_V  |->  ( iota_ v  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v ) ) )  |`  { x  e.  On  |  E. t  e.  A  A. z  e.  (recs ( ( h  e. 
_V  |->  ( iota_ v  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v ) ) )
" x ) z R t } ) ,  (/) )
4 iffalse 3746 . 2  |-  ( -.  ( R  We  A  /\  R Se  A )  ->  if ( ( R  We  A  /\  R Se  A ) ,  (recs ( ( h  e. 
_V  |->  ( iota_ v  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v ) ) )  |`  { x  e.  On  |  E. t  e.  A  A. z  e.  (recs ( ( h  e. 
_V  |->  ( iota_ v  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e. 
{ w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v ) ) )
" x ) z R t } ) ,  (/) )  =  (/) )
53, 4syl5eq 2480 1  |-  ( -.  ( R  We  A  /\  R Se  A )  ->  F  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652   A.wral 2705   E.wrex 2706   {crab 2709   _Vcvv 2956   (/)c0 3628   ifcif 3739   class class class wbr 4212    e. cmpt 4266   Se wse 4539    We wwe 4540   Oncon0 4581   ran crn 4879    |` cres 4880   "cima 4881   iota_crio 6542  recscrecs 6632  OrdIsocoi 7478
This theorem is referenced by:  oicl  7498  oif  7499  oiexg  7504
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-if 3740  df-oi 7479
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