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	⊢ ℵ Isom E, E (On, {x∣(ω ⊆ x ∧ (card ‘x) = x)})

Proof of Theorem alephiso

Step	Hyp	Ref	Expression
1		alephfnon 3668	. . . . . . . . . 10 ⊢ ℵ Fn On
2		isinfcard 3692	. . . . . . . . . . . 12 ⊢ ((ω ⊆ x ∧ (card ‘x) = x) ↔ x ∈ ran ℵ)
3	2	bicomi 150	. . . . . . . . . . 11 ⊢ (x ∈ ran ℵ ↔ (ω ⊆ x ∧ (card ‘x) = x))
4	3	biabri 1180	. . . . . . . . . 10 ⊢ ran ℵ = {x∣(ω ⊆ x ∧ (card ‘x) = x)}
5	1, 4	pm3.2i 234	. . . . . . . . 9 ⊢ (ℵ Fn On ∧ ran ℵ = {x∣(ω ⊆ x ∧ (card ‘x) = x)})
6		df-fo 2436	. . . . . . . . 9 ⊢ (ℵ:On–onto→{x∣(ω ⊆ x ∧ (card ‘x) = x)} ↔ (ℵ Fn On ∧ ran ℵ = {x∣(ω ⊆ x ∧ (card ‘x) = x)}))
7	5, 6	mpbir 165	. . . . . . . 8 ⊢ ℵ:On–onto→{x∣(ω ⊆ x ∧ (card ‘x) = x)}
8		fof 2788	. . . . . . . 8 ⊢ (ℵ:On–onto→{x∣(ω ⊆ x ∧ (card ‘x) = x)} → ℵ:On–→{x∣(ω ⊆ x ∧ (card ‘x) = x)})
9	7, 8	ax-mp 6	. . . . . . 7 ⊢ ℵ:On–→{x∣(ω ⊆ x ∧ (card ‘x) = x)}
10		aleph11 3684	. . . . . . . . 9 ⊢ ((y ∈ On ∧ z ∈ On) → ((ℵ ‘y) = (ℵ ‘z) ↔ y = z))
11	10	biimpd 135	. . . . . . . 8 ⊢ ((y ∈ On ∧ z ∈ On) → ((ℵ ‘y) = (ℵ ‘z) → y = z))
12	11	rgen2 1248	. . . . . . 7 ⊢ ∀y ∈ On ∀z ∈ On ((ℵ ‘y) = (ℵ ‘z) → y = z)
13	9, 12	pm3.2i 234	. . . . . 6 ⊢ (ℵ:On–→{x∣(ω ⊆ x ∧ (card ‘x) = x)} ∧ ∀y ∈ On ∀z ∈ On ((ℵ ‘y) = (ℵ ‘z) → y = z))
14		f1fv 2916	. . . . . 6 ⊢ (ℵ:On–1-1→{x∣(ω ⊆ x ∧ (card ‘x) = x)} ↔ (ℵ:On–→{x∣(ω ⊆ x ∧ (card ‘x) = x)} ∧ ∀y ∈ On ∀z ∈ On ((ℵ ‘y) = (ℵ ‘z) → y = z)))
15	13, 14	mpbir 165	. . . . 5 ⊢ ℵ:On–1-1→{x∣(ω ⊆ x ∧ (card ‘x) = x)}
16	15, 7	pm3.2i 234	. . . 4 ⊢ (ℵ:On–1-1→{x∣(ω ⊆ x ∧ (card ‘x) = x)} ∧ ℵ:On–onto→{x∣(ω ⊆ x ∧ (card ‘x) = x)})
17		df-f1o 2437	. . . 4 ⊢ (ℵ:On–1-1-onto→{x∣(ω ⊆ x ∧ (card ‘x) = x)} ↔ (ℵ:On–1-1→{x∣(ω ⊆ x ∧ (card ‘x) = x)} ∧ ℵ:On–onto→{x∣(ω ⊆ x ∧ (card ‘x) = x)}))
18	16, 17	mpbir 165	. . 3 ⊢ ℵ:On–1-1-onto→{x∣(ω ⊆ x ∧ (card ‘x) = x)}
19		alephord2 3681	. . . . 5 ⊢ ((y ∈ On ∧ z ∈ On) → (y ∈ z ↔ (ℵ ‘y) ∈ (ℵ ‘z)))
20		epel 2124	. . . . 5 ⊢ (yEz ↔ y ∈ z)
21		fvex 2838	. . . . . 6 ⊢ (ℵ ‘y) ∈ V
22		fvex 2838	. . . . . 6 ⊢ (ℵ ‘z) ∈ V
23	21, 22	epelc 2123	. . . . 5 ⊢ ((ℵ ‘y)E(ℵ ‘z) ↔ (ℵ ‘y) ∈ (ℵ ‘z))
24	19, 20, 23	3bitr4g 428	. . . 4 ⊢ ((y ∈ On ∧ z ∈ On) → (yEz ↔ (ℵ ‘y)E(ℵ ‘z)))
25	24	rgen2 1248	. . 3 ⊢ ∀y ∈ On ∀z ∈ On (yEz ↔ (ℵ ‘y)E(ℵ ‘z))
26	18, 25	pm3.2i 234	. 2 ⊢ (ℵ:On–1-1-onto→{x∣(ω ⊆ x ∧ (card ‘x) = x)} ∧ ∀y ∈ On ∀z ∈ On (yEz ↔ (ℵ ‘y)E(ℵ ‘z)))
27		df-iso 2439	. 2 ⊢ (ℵ Isom E, E (On, {x∣(ω ⊆ x ∧ (card ‘x) = x)}) ↔ (ℵ:On–1-1-onto→{x∣(ω ⊆ x ∧ (card ‘x) = x)} ∧ ∀y ∈ On ∀z ∈ On (yEz ↔ (ℵ ‘y)E(ℵ ‘z))))
28	26, 27	mpbir 165	1 ⊢ ℵ Isom E, E (On, {x∣(ω ⊆ x ∧ (card ‘x) = x)})

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = weq 797   ∈ wel 803  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201   ⊆ wss 1487   class class class wbr 2054  Ecep 2056  Oncon0 2199  ωcom 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  ℵcale 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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