Theorem ordiso 7327

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 5083	. . . . 5 |- ( A e. On -> ( _I |` A ) e. _V )
2		isoid 5983	. . . . 5 |- ( _I |` A ) Isom _E , _E ( A , A )
3		isoeq1 5974	. . . . . 6 |- ( f = ( _I |` A ) -> ( f Isom _E , _E ( A , A ) <-> ( _I |` A ) Isom _E , _E ( A , A ) ) )
4	3	spcegv 2905	. . . . 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 1492	. . . 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 5978	. . . 4 |- ( A = B -> ( f Isom _E , _E ( A , A ) <-> f Isom _E , _E ( A , B ) ) )
8	7	exbidv 1874	. . 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 4496	. . . 4 |- ( A e. On -> Ord A )
11		eloni 4496	. . . 4 |- ( B e. On -> Ord B )
12		ordiso2 7326	. . . . . 6 |- ( ( f Isom _E , _E ( A , B ) /\ Ord A /\ Ord B ) -> A = B )
13	12	3coml 1237	. . . . 5 |- ( ( Ord A /\ Ord B /\ f Isom _E , _E ( A , B ) ) -> A = B )
14	13	3expia 1232	. . . 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 1868	. 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 1398   E.wex 1541   e. wcel 2203   _Vcvv 2813   _E cep 4408   _I cid 4409   Ord word 4483   Oncon0 4484   |` cres 4751   Isom wiso 5353

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-iord 4487  df-on 4489  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361

This theorem is referenced by: (None)