Theorem alephiso 3697

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		alephfnon 3668	. . . . . . . . . 10 |- aleph Fn On
2		isinfcard 3692	. . . . . . . . . . . 12 |- ((om (_ x /\ (card` x) = x) <-> x e. ran aleph)
3	2	bicomi 150	. . . . . . . . . . 11 |- (x e. ran aleph <-> (om (_ x /\ (card` x) = x))
4	3	biabri 1180	. . . . . . . . . 10 |- ran aleph = {x | (om (_ x /\ (card` x) = x)}
5	1, 4	pm3.2i 234	. . . . . . . . 9 |- (aleph Fn On /\ ran aleph = {x | (om (_ x /\ (card` x) = x)})
6		df-fo 2436	. . . . . . . . 9 |- (aleph:On-onto->{x | (om (_ x /\ (card` x) = x)} <-> (aleph Fn On /\ ran aleph = {x | (om (_ x /\ (card` x) = x)}))
7	5, 6	mpbir 165	. . . . . . . 8 |- aleph:On-onto->{x | (om (_ x /\ (card` x) = x)}
8		fof 2788	. . . . . . . 8 |- (aleph:On-onto->{x | (om (_ x /\ (card` x) = x)} -> aleph:On-->{x | (om (_ x /\ (card` x) = x)})
9	7, 8	ax-mp 6	. . . . . . 7 |- aleph:On-->{x | (om (_ x /\ (card` x) = x)}
10		aleph11 3684	. . . . . . . . 9 |- ((y e. On /\ z e. On) -> ((aleph` y) = (aleph` z) <-> y = z))
11	10	biimpd 135	. . . . . . . 8 |- ((y e. On /\ z e. On) -> ((aleph` y) = (aleph` z) -> y = z))
12	11	rgen2 1248	. . . . . . 7 |- A.y e. On A.z e. On ((aleph` y) = (aleph` z) -> y = z)
13	9, 12	pm3.2i 234	. . . . . 6 |- (aleph:On-->{x | (om (_ x /\ (card` x) = x)} /\ A.y e. On A.z e. On ((aleph` y) = (aleph` z) -> y = z))
14		f1fv 2916	. . . . . 6 |- (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)))
15	13, 14	mpbir 165	. . . . 5 |- aleph:On-1-1->{x | (om (_ x /\ (card` x) = x)}
16	15, 7	pm3.2i 234	. . . 4 |- (aleph:On-1-1->{x | (om (_ x /\ (card` x) = x)} /\ aleph:On-onto->{x | (om (_ x /\ (card` x) = x)})
17		df-f1o 2437	. . . 4 |- (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)}))
18	16, 17	mpbir 165	. . 3 |- aleph:On-1-1-onto->{x | (om (_ x /\ (card` x) = x)}
19		alephord2 3681	. . . . 5 |- ((y e. On /\ z e. On) -> (y e. z <-> (aleph` y) e. (aleph` z)))
20		epel 2124	. . . . 5 |- (yEz <-> y e. z)
21		fvex 2838	. . . . . 6 |- (aleph` y) e. V
22		fvex 2838	. . . . . 6 |- (aleph` z) e. V
23	21, 22	epelc 2123	. . . . 5 |- ((aleph` y)E(aleph` z) <-> (aleph` y) e. (aleph` z))
24	19, 20, 23	3bitr4g 428	. . . 4 |- ((y e. On /\ z e. On) -> (yEz <-> (aleph` y)E(aleph` z)))
25	24	rgen2 1248	. . 3 |- A.y e. On A.z e. On (yEz <-> (aleph` y)E(aleph` z))
26	18, 25	pm3.2i 234	. 2 |- (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)))
27		df-iso 2439	. 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))))
28	26, 27	mpbir 165	1 |- aleph Isom E, E (On, {x | (om (_ x /\ (card` x) = x)})

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = weq 797   e. wel 803  {cab 1090   = wceq 1091   e. wcel 1092  A.wral 1201   (_ wss 1487   class class class wbr 2054  Ecep 2056  Oncon0 2199  omcom 2372  ran crn 2411   Fn wfn 2417  -->wf 2418  -1-1->wf1 2419  -onto->wfo 2420  -1-1-onto->wf1o 2421  ` cfv 2422   Isom wiso 2423  cardccrd 3620  alephcale 3621

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-iso 2439  df-rdg 2970  df-er 3200  df-en 3274  df-dom 3275  df-sdom 3276  df-card 3623  df-aleph 3624

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