Theorem alephiso 4864

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.

Assertion

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

Proof of Theorem alephiso

Step	Hyp	Ref	Expression
1		df-iso 3189	. 2 |- (aleph Isom E, E (On, {x | (om (_ x /\ (card` x) = x)}) <-> (aleph:On-1-1-onto->{x | (om (_ x /\ (card` x) = x)} /\ A.y e. On A.z e. On (yEz <-> (aleph` y)E(aleph` z))))
2		df-f1o 3187	. . 3 |- (aleph:On-1-1-onto->{x | (om (_ x /\ (card` x) = x)} <-> (aleph:On-1-1->{x | (om (_ x /\ (card` x) = x)} /\ aleph:On-onto->{x | (om (_ x /\ (card` x) = x)}))
3		f1fv 3859	. . . 4 |- (aleph:On-1-1->{x | (om (_ x /\ (card` x) = x)} <-> (aleph:On-->{x | (om (_ x /\ (card` x) = x)} /\ A.y e. On A.z e. On ((aleph` y) = (aleph` z) -> y = z)))
4		df-fo 3186	. . . . . 6 |- (aleph:On-onto->{x | (om (_ x /\ (card` x) = x)} <-> (aleph Fn On /\ ran aleph = {x | (om (_ x /\ (card` x) = x)}))
5		alephfnon 4834	. . . . . 6 |- aleph Fn On
6		isinfcard 4859	. . . . . . . 8 |- ((om (_ x /\ (card` x) = x) <-> x e. ran aleph)
7	6	bicomi 172	. . . . . . 7 |- (x e. ran aleph <-> (om (_ x /\ (card` x) = x))
8	7	abbi2i 1566	. . . . . 6 |- ran aleph = {x | (om (_ x /\ (card` x) = x)}
9	4, 5, 8	mpbir2an 728	. . . . 5 |- aleph:On-onto->{x | (om (_ x /\ (card` x) = x)}
10		fof 3657	. . . . 5 |- (aleph:On-onto->{x | (om (_ x /\ (card` x) = x)} -> aleph:On-->{x | (om (_ x /\ (card` x) = x)})
11	9, 10	ax-mp 7	. . . 4 |- aleph:On-->{x | (om (_ x /\ (card` x) = x)}
12		aleph11 4851	. . . . . 6 |- ((y e. On /\ z e. On) -> ((aleph` y) = (aleph` z) <-> y = z))
13	12	biimpd 153	. . . . 5 |- ((y e. On /\ z e. On) -> ((aleph` y) = (aleph` z) -> y = z))
14	13	rgen2a 1691	. . . 4 |- A.y e. On A.z e. On ((aleph` y) = (aleph` z) -> y = z)
15	3, 11, 14	mpbir2an 728	. . 3 |- aleph:On-1-1->{x | (om (_ x /\ (card` x) = x)}
16	2, 15, 9	mpbir2an 728	. 2 |- aleph:On-1-1-onto->{x | (om (_ x /\ (card` x) = x)}
17		alephord2 4848	. . . 4 |- ((y e. On /\ z e. On) -> (y e. z <-> (aleph` y) e. (aleph` z)))
18		epel 2823	. . . 4 |- (yEz <-> y e. z)
19		fvex 3717	. . . . 5 |- (aleph` y) e. V
20		fvex 3717	. . . . 5 |- (aleph` z) e. V
21	19, 20	epelc 2822	. . . 4 |- ((aleph` y)E(aleph` z) <-> (aleph` y) e. (aleph` z))
22	17, 18, 21	3bitr4g 553	. . 3 |- ((y e. On /\ z e. On) -> (yEz <-> (aleph` y)E(aleph` z)))
23	22	rgen2a 1691	. 2 |- A.y e. On A.z e. On (yEz <-> (aleph` y)E(aleph` z))
24	1, 16, 23	mpbir2an 728	1 |- aleph Isom E, E (On, {x | (om (_ x /\ (card` x) = x)})

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  {cab 1456  A.wral 1637   (_ wss 2037   class class class wbr 2609  Ecep 2819  Oncon0 2938  omcom 3121  ran crn 3161   Fn wfn 3167  -->wf 3168  -1-1->wf1 3169  -onto->wfo 3170  -1-1-onto->wf1o 3171  ` cfv 3172   Isom wiso 3173  cardccrd 4785  alephcale 4786

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-reg 4565  ax-inf2 4597  ax-ac 4716

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-iso 3189  df-rdg 3917  df-er 4245  df-en 4351  df-dom 4352  df-sdom 4353  df-card 4788  df-aleph 4789

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