Theorem alephiso 10015

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	⊢ ℵ Isom E , E (On, {𝑥 ∣ (ω ⊆ 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 9982	. . . . . 6 ⊢ ℵ Fn On
2		isinfcard 10009	. . . . . . . 8 ⊢ ((ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥) ↔ 𝑥 ∈ ran ℵ)
3	2	bicomi 226	. . . . . . 7 ⊢ (𝑥 ∈ ran ℵ ↔ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥))
4	3	eqabi 2876	. . . . . 6 ⊢ ran ℵ = {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)}
5		df-fo 6494	. . . . . 6 ⊢ (ℵ:On–onto→{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} ↔ (ℵ Fn On ∧ ran ℵ = {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)}))
6	1, 4, 5	mpbir2an 718	. . . . 5 ⊢ ℵ:On–onto→{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)}
7		fof 6742	. . . . 5 ⊢ (ℵ:On–onto→{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} → ℵ:On⟶{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)})
8	6, 7	ax-mp 5	. . . 4 ⊢ ℵ:On⟶{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)}
9		aleph11 10001	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → ((ℵ‘𝑦) = (ℵ‘𝑧) ↔ 𝑦 = 𝑧))
10	9	biimpd 231	. . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → ((ℵ‘𝑦) = (ℵ‘𝑧) → 𝑦 = 𝑧))
11	10	rgen2 3181	. . . 4 ⊢ ∀𝑦 ∈ On ∀𝑧 ∈ On ((ℵ‘𝑦) = (ℵ‘𝑧) → 𝑦 = 𝑧)
12		dff13 7201	. . . 4 ⊢ (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)))
13	8, 11, 12	mpbir2an 718	. . 3 ⊢ ℵ:On–1-1→{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)}
14		df-f1o 6495	. . 3 ⊢ (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)}))
15	13, 6, 14	mpbir2an 718	. 2 ⊢ ℵ:On–1-1-onto→{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)}
16		alephord2 9993	. . . 4 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 ∈ 𝑧 ↔ (ℵ‘𝑦) ∈ (ℵ‘𝑧)))
17		epel 5523	. . . 4 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧)
18		fvex 6843	. . . . 5 ⊢ (ℵ‘𝑧) ∈ V
19	18	epeli 5522	. . . 4 ⊢ ((ℵ‘𝑦) E (ℵ‘𝑧) ↔ (ℵ‘𝑦) ∈ (ℵ‘𝑧))
20	16, 17, 19	3bitr4g 316	. . 3 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 E 𝑧 ↔ (ℵ‘𝑦) E (ℵ‘𝑧)))
21	20	rgen2 3181	. 2 ⊢ ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑦 E 𝑧 ↔ (ℵ‘𝑦) E (ℵ‘𝑧))
22		df-isom 6497	. 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 (𝑦 E 𝑧 ↔ (ℵ‘𝑦) E (ℵ‘𝑧))))
23	15, 21, 22	mpbir2an 718	1 ⊢ ℵ Isom E , E (On, {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  {cab 2719  ∀wral 3055   ⊆ wss 3884   class class class wbr 5074   E cep 5519  ran crn 5621  Oncon0 6313   Fn wfn 6483  ⟶wf 6484  –1-1→wf1 6485  –onto→wfo 6486  –1-1-onto→wf1o 6487  ‘cfv 6488   Isom wiso 6489  ωcom 7809  cardccrd 9854  ℵcale 9855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681  ax-inf2 9557

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3904  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-se 5574  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-isom 6497  df-riota 7316  df-ov 7362  df-om 7810  df-2nd 7934  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8343  df-1o 8399  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-oi 9419  df-har 9466  df-card 9858  df-aleph 9859

This theorem is referenced by:  alephiso2  44015