Theorem ordiso 7164

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 5023	. . . . 5 |- ( A e. On -> ( _I |` A ) e. _V )
2		isoid 5902	. . . . 5 |- ( _I |` A ) Isom _E , _E ( A , A )
3		isoeq1 5893	. . . . . 6 |- ( f = ( _I |` A ) -> ( f Isom _E , _E ( A , A ) <-> ( _I |` A ) Isom _E , _E ( A , A ) ) )
4	3	spcegv 2868	. . . . 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 1467	. . . 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 5897	. . . 4 |- ( A = B -> ( f Isom _E , _E ( A , A ) <-> f Isom _E , _E ( A , B ) ) )
8	7	exbidv 1849	. . 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 4440	. . . 4 |- ( A e. On -> Ord A )
11		eloni 4440	. . . 4 |- ( B e. On -> Ord B )
12		ordiso2 7163	. . . . . 6 |- ( ( f Isom _E , _E ( A , B ) /\ Ord A /\ Ord B ) -> A = B )
13	12	3coml 1213	. . . . 5 |- ( ( Ord A /\ Ord B /\ f Isom _E , _E ( A , B ) ) -> A = B )
14	13	3expia 1208	. . . 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 1843	. 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 1373   E.wex 1516   e. wcel 2178   _Vcvv 2776   _E cep 4352   _I cid 4353   Ord word 4427   Oncon0 4428   |` cres 4695   Isom wiso 5291

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-eprel 4354  df-id 4358  df-iord 4431  df-on 4433  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-isom 5299

This theorem is referenced by: (None)