Theorem ordiso 7066

Description: Order-isomorphic ordinal numbers are equal. (Contributed by Jeff Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro, 24-Jun-2015.)

Assertion

Ref	Expression
ordiso	|- ( ( A e. On /\ B e. On ) -> ( A = B <-> E. f f Isom _E , _E ( A , B ) ) )

Distinct variable groups:   A, f   B, f

Proof of Theorem ordiso

Step	Hyp	Ref	Expression
1		resiexg 4970	. . . . 5 |- ( A e. On -> ( _I |` A ) e. _V )
2		isoid 5832	. . . . 5 |- ( _I |` A ) Isom _E , _E ( A , A )
3		isoeq1 5823	. . . . . 6 |- ( f = ( _I |` A ) -> ( f Isom _E , _E ( A , A ) <-> ( _I |` A ) Isom _E , _E ( A , A ) ) )
4	3	spcegv 2840	. . . . 5 |- ( ( _I |` A ) e. _V -> ( ( _I |` A ) Isom _E , _E ( A , A ) -> E. f f Isom _E , _E ( A , A ) ) )
5	1, 2, 4	mpisyl 1457	. . . 4 |- ( A e. On -> E. f f Isom _E , _E ( A , A ) )
6	5	adantr 276	. . 3 |- ( ( A e. On /\ B e. On ) -> E. f f Isom _E , _E ( A , A ) )
7		isoeq5 5827	. . . 4 |- ( A = B -> ( f Isom _E , _E ( A , A ) <-> f Isom _E , _E ( A , B ) ) )
8	7	exbidv 1836	. . 3 |- ( A = B -> ( E. f f Isom _E , _E ( A , A ) <-> E. f f Isom _E , _E ( A , B ) ) )
9	6, 8	syl5ibcom 155	. 2 |- ( ( A e. On /\ B e. On ) -> ( A = B -> E. f f Isom _E , _E ( A , B ) ) )
10		eloni 4393	. . . 4 |- ( A e. On -> Ord A )
11		eloni 4393	. . . 4 |- ( B e. On -> Ord B )
12		ordiso2 7065	. . . . . 6 |- ( ( f Isom _E , _E ( A , B ) /\ Ord A /\ Ord B ) -> A = B )
13	12	3coml 1212	. . . . 5 |- ( ( Ord A /\ Ord B /\ f Isom _E , _E ( A , B ) ) -> A = B )
14	13	3expia 1207	. . . 4 |- ( ( Ord A /\ Ord B ) -> ( f Isom _E , _E ( A , B ) -> A = B ) )
15	10, 11, 14	syl2an 289	. . 3 |- ( ( A e. On /\ B e. On ) -> ( f Isom _E , _E ( A , B ) -> A = B ) )
16	15	exlimdv 1830	. 2 |- ( ( A e. On /\ B e. On ) -> ( E. f f Isom _E , _E ( A , B ) -> A = B ) )
17	9, 16	impbid 129	1 |- ( ( A e. On /\ B e. On ) -> ( A = B <-> E. f f Isom _E , _E ( A , B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1503   e. wcel 2160   _Vcvv 2752   _E cep 4305   _I cid 4306   Ord word 4380   Oncon0 4381   |` cres 4646   Isom wiso 5236

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-iord 4384  df-on 4386  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244

This theorem is referenced by: (None)