Theorem alephiso 8921

Description: Aleph is an order isomorphism of the class of ordinal numbers onto the class of infinite cardinals. Definition 10.27 of [TakeutiZaring] p. 90. (Contributed by NM, 3-Aug-2004.)

Assertion

Ref	Expression
alephiso	|- aleph Isom _E , _E ( On , { x | ( om C_ x /\ ( card ` x ) = x ) } )

Proof of Theorem alephiso

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		alephfnon 8888	. . . . . 6 |- aleph Fn On
2		isinfcard 8915	. . . . . . . 8 |- ( ( om C_ x /\ ( card ` x ) = x ) <-> x e. ran aleph )
3	2	bicomi 214	. . . . . . 7 |- ( x e. ran aleph <-> ( om C_ x /\ ( card ` x ) = x ) )
4	3	abbi2i 2738	. . . . . 6 |- ran aleph = { x | ( om C_ x /\ ( card ` x ) = x ) }
5		df-fo 5894	. . . . . 6 |- ( aleph : On -onto-> { x | ( om C_ x /\ ( card ` x ) = x ) } <-> ( aleph Fn On /\ ran aleph = { x | ( om C_ x /\ ( card ` x ) = x ) } ) )
6	1, 4, 5	mpbir2an 955	. . . . 5 |- aleph : On -onto-> { x | ( om C_ x /\ ( card ` x ) = x ) }
7		fof 6115	. . . . 5 |- ( aleph : On -onto-> { x | ( om C_ x /\ ( card ` x ) = x ) } -> aleph : On --> { x | ( om C_ x /\ ( card ` x ) = x ) } )
8	6, 7	ax-mp 5	. . . 4 |- aleph : On --> { x | ( om C_ x /\ ( card ` x ) = x ) }
9		aleph11 8907	. . . . . 6 |- ( ( y e. On /\ z e. On ) -> ( ( aleph ` y ) = ( aleph ` z ) <-> y = z ) )
10	9	biimpd 219	. . . . 5 |- ( ( y e. On /\ z e. On ) -> ( ( aleph ` y ) = ( aleph ` z ) -> y = z ) )
11	10	rgen2a 2977	. . . 4 |- A. y e. On A. z e. On ( ( aleph ` y ) = ( aleph ` z ) -> y = z )
12		dff13 6512	. . . 4 |- ( aleph : On -1-1-> { x | ( om C_ x /\ ( card ` x ) = x ) } <-> ( aleph : On --> { x | ( om C_ x /\ ( card ` x ) = x ) } /\ A. y e. On A. z e. On ( ( aleph ` y ) = ( aleph ` z ) -> y = z ) ) )
13	8, 11, 12	mpbir2an 955	. . 3 |- aleph : On -1-1-> { x | ( om C_ x /\ ( card ` x ) = x ) }
14		df-f1o 5895	. . 3 |- ( aleph : On -1-1-onto-> { x | ( om C_ x /\ ( card ` x ) = x ) } <-> ( aleph : On -1-1-> { x | ( om C_ x /\ ( card ` x ) = x ) } /\ aleph : On -onto-> { x | ( om C_ x /\ ( card ` x ) = x ) } ) )
15	13, 6, 14	mpbir2an 955	. 2 |- aleph : On -1-1-onto-> { x | ( om C_ x /\ ( card ` x ) = x ) }
16		alephord2 8899	. . . 4 |- ( ( y e. On /\ z e. On ) -> ( y e. z <-> ( aleph ` y ) e. ( aleph ` z ) ) )
17		epel 5032	. . . 4 |- ( y _E z <-> y e. z )
18		fvex 6201	. . . . 5 |- ( aleph ` z ) e. _V
19	18	epelc 5031	. . . 4 |- ( ( aleph ` y ) _E ( aleph ` z ) <-> ( aleph ` y ) e. ( aleph ` z ) )
20	16, 17, 19	3bitr4g 303	. . 3 |- ( ( y e. On /\ z e. On ) -> ( y _E z <-> ( aleph ` y ) _E ( aleph ` z ) ) )
21	20	rgen2a 2977	. 2 |- A. y e. On A. z e. On ( y _E z <-> ( aleph ` y ) _E ( aleph ` z ) )
22		df-isom 5897	. 2 |- ( aleph Isom _E , _E ( On , { x | ( om C_ x /\ ( card ` x ) = x ) } ) <-> ( aleph : On -1-1-onto-> { x | ( om C_ x /\ ( card ` x ) = x ) } /\ A. y e. On A. z e. On ( y _E z <-> ( aleph ` y ) _E ( aleph ` z ) ) ) )
23	15, 21, 22	mpbir2an 955	1 |- aleph Isom _E , _E ( On , { x | ( om C_ x /\ ( card ` x ) = x ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   C_ wss 3574   class class class wbr 4653   _E cep 5028   ran crn 5115   Oncon0 5723   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889   omcom 7065   cardccrd 8761   alephcale 8762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-har 8463  df-card 8765  df-aleph 8766

This theorem is referenced by: (None)