Theorem alephiso 4892

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 3199	. 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 3197	. . 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 3874	. . . 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 3196	. . . . . 6 |- (aleph:On-onto->{x | (om (_ x /\ (card` x) = x)} <-> (aleph Fn On /\ ran aleph = {x | (om (_ x /\ (card` x) = x)}))
5		alephfnon 4862	. . . . . 6 |- aleph Fn On
6		isinfcard 4887	. . . . . . . 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 1574	. . . . . 6 |- ran aleph = {x | (om (_ x /\ (card` x) = x)}
9	4, 5, 8	mpbir2an 730	. . . . 5 |- aleph:On-onto->{x | (om (_ x /\ (card` x) = x)}
10		fof 3672	. . . . 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 4879	. . . . . 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 1699	. . . 4 |- A.y e. On A.z e. On ((aleph` y) = (aleph` z) -> y = z)
15	3, 11, 14	mpbir2an 730	. . 3 |- aleph:On-1-1->{x | (om (_ x /\ (card` x) = x)}
16	2, 15, 9	mpbir2an 730	. 2 |- aleph:On-1-1-onto->{x | (om (_ x /\ (card` x) = x)}
17		alephord2 4876	. . . 4 |- ((y e. On /\ z e. On) -> (y e. z <-> (aleph` y) e. (aleph` z)))
18		epel 2834	. . . 4 |- (yEz <-> y e. z)
19		fvex 3732	. . . . 5 |- (aleph` y) e. V
20		fvex 3732	. . . . 5 |- (aleph` z) e. V
21	19, 20	epelc 2833	. . . 4 |- ((aleph` y)E(aleph` z) <-> (aleph` y) e. (aleph` z))
22	17, 18, 21	3bitr4g 555	. . 3 |- ((y e. On /\ z e. On) -> (yEz <-> (aleph` y)E(aleph` z)))
23	22	rgen2a 1699	. 2 |- A.y e. On A.z e. On (yEz <-> (aleph` y)E(aleph` z))
24	1, 16, 23	mpbir2an 730	1 |- aleph Isom E, E (On, {x | (om (_ x /\ (card` x) = x)})

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  A.wral 1645   (_ wss 2047   class class class wbr 2619  Ecep 2830  Oncon0 2948  omcom 3131  ran crn 3171   Fn wfn 3177  -->wf 3178  -1-1->wf1 3179  -onto->wfo 3180  -1-1-onto->wf1o 3181  ` cfv 3182   Isom wiso 3183  cardccrd 4813  alephcale 4814

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-reg 4593  ax-inf2 4625  ax-ac 4744

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-iso 3199  df-rdg 3932  df-er 4261  df-en 4368  df-dom 4369  df-sdom 4370  df-card 4816  df-aleph 4817

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