Theorem ordiso 7097

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 4988	. . . . 5 |- ( A e. On -> ( _I |` A ) e. _V )
2		isoid 5854	. . . . 5 |- ( _I |` A ) Isom _E , _E ( A , A )
3		isoeq1 5845	. . . . . 6 |- ( f = ( _I |` A ) -> ( f Isom _E , _E ( A , A ) <-> ( _I |` A ) Isom _E , _E ( A , A ) ) )
4	3	spcegv 2849	. . . . 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 5849	. . . 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 4407	. . . 4 |- ( A e. On -> Ord A )
11		eloni 4407	. . . 4 |- ( B e. On -> Ord B )
12		ordiso2 7096	. . . . . 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 2164   _Vcvv 2760   _E cep 4319   _I cid 4320   Ord word 4394   Oncon0 4395   |` cres 4662   Isom wiso 5256

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-eprel 4321  df-id 4325  df-iord 4398  df-on 4400  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264

This theorem is referenced by: (None)