Theorem ordiso 7001

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 4929	. . . . 5 |- ( A e. On -> ( _I |` A ) e. _V )
2		isoid 5778	. . . . 5 |- ( _I |` A ) Isom _E , _E ( A , A )
3		isoeq1 5769	. . . . . 6 |- ( f = ( _I |` A ) -> ( f Isom _E , _E ( A , A ) <-> ( _I |` A ) Isom _E , _E ( A , A ) ) )
4	3	spcegv 2814	. . . . 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 1434	. . . 4 |- ( A e. On -> E. f f Isom _E , _E ( A , A ) )
6	5	adantr 274	. . 3 |- ( ( A e. On /\ B e. On ) -> E. f f Isom _E , _E ( A , A ) )
7		isoeq5 5773	. . . 4 |- ( A = B -> ( f Isom _E , _E ( A , A ) <-> f Isom _E , _E ( A , B ) ) )
8	7	exbidv 1813	. . 3 |- ( A = B -> ( E. f f Isom _E , _E ( A , A ) <-> E. f f Isom _E , _E ( A , B ) ) )
9	6, 8	syl5ibcom 154	. 2 |- ( ( A e. On /\ B e. On ) -> ( A = B -> E. f f Isom _E , _E ( A , B ) ) )
10		eloni 4353	. . . 4 |- ( A e. On -> Ord A )
11		eloni 4353	. . . 4 |- ( B e. On -> Ord B )
12		ordiso2 7000	. . . . . 6 |- ( ( f Isom _E , _E ( A , B ) /\ Ord A /\ Ord B ) -> A = B )
13	12	3coml 1200	. . . . 5 |- ( ( Ord A /\ Ord B /\ f Isom _E , _E ( A , B ) ) -> A = B )
14	13	3expia 1195	. . . 4 |- ( ( Ord A /\ Ord B ) -> ( f Isom _E , _E ( A , B ) -> A = B ) )
15	10, 11, 14	syl2an 287	. . 3 |- ( ( A e. On /\ B e. On ) -> ( f Isom _E , _E ( A , B ) -> A = B ) )
16	15	exlimdv 1807	. 2 |- ( ( A e. On /\ B e. On ) -> ( E. f f Isom _E , _E ( A , B ) -> A = B ) )
17	9, 16	impbid 128	1 |- ( ( A e. On /\ B e. On ) -> ( A = B <-> E. f f Isom _E , _E ( A , B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   E.wex 1480   e. wcel 2136   _Vcvv 2726   _E cep 4265   _I cid 4266   Ord word 4340   Oncon0 4341   |` cres 4606   Isom wiso 5189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-iord 4344  df-on 4346  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197

This theorem is referenced by: (None)