Theorem oi0 8433

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.

Step	Hyp	Ref	Expression
1		oicl.1	. . 3 |- F = OrdIso ( R , A )
2		df-oi 8415	. . 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 } ) , (/) )
3	1, 2	eqtri 2644	. 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 4095	. 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 } ) , (/) ) = (/) )
5	3, 4	syl5eq 2668	1 |- ( -. ( R We A /\ R Se A ) -> F = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   (/)c0 3915   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   Se wse 5071   We wwe 5072   ran crn 5115   |` cres 5116   "cima 5117   Oncon0 5723   iota_crio 6610  recscrecs 7467  OrdIsocoi 8414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-if 4087  df-oi 8415

This theorem is referenced by:  oicl  8434  oif  8435  oiexg  8440