Theorem alephiso 9509

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 9476	. . . . . 6 ⊢ ℵ Fn On
2		isinfcard 9503	. . . . . . . 8 ⊢ ((ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥) ↔ 𝑥 ∈ ran ℵ)
3	2	bicomi 227	. . . . . . 7 ⊢ (𝑥 ∈ ran ℵ ↔ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥))
4	3	abbi2i 2929	. . . . . 6 ⊢ ran ℵ = {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)}
5		df-fo 6330	. . . . . 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 710	. . . . 5 ⊢ ℵ:On–onto→{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)}
7		fof 6565	. . . . 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 9495	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → ((ℵ‘𝑦) = (ℵ‘𝑧) ↔ 𝑦 = 𝑧))
10	9	biimpd 232	. . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → ((ℵ‘𝑦) = (ℵ‘𝑧) → 𝑦 = 𝑧))
11	10	rgen2 3168	. . . 4 ⊢ ∀𝑦 ∈ On ∀𝑧 ∈ On ((ℵ‘𝑦) = (ℵ‘𝑧) → 𝑦 = 𝑧)
12		dff13 6991	. . . 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 710	. . 3 ⊢ ℵ:On–1-1→{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)}
14		df-f1o 6331	. . 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 710	. 2 ⊢ ℵ:On–1-1-onto→{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)}
16		alephord2 9487	. . . 4 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 ∈ 𝑧 ↔ (ℵ‘𝑦) ∈ (ℵ‘𝑧)))
17		epel 5433	. . . 4 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧)
18		fvex 6658	. . . . 5 ⊢ (ℵ‘𝑧) ∈ V
19	18	epeli 5432	. . . 4 ⊢ ((ℵ‘𝑦) E (ℵ‘𝑧) ↔ (ℵ‘𝑦) ∈ (ℵ‘𝑧))
20	16, 17, 19	3bitr4g 317	. . 3 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 E 𝑧 ↔ (ℵ‘𝑦) E (ℵ‘𝑧)))
21	20	rgen2 3168	. 2 ⊢ ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑦 E 𝑧 ↔ (ℵ‘𝑦) E (ℵ‘𝑧))
22		df-isom 6333	. 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 710	1 ⊢ ℵ Isom E , E (On, {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2776  ∀wral 3106   ⊆ wss 3881   class class class wbr 5030   E cep 5429  ran crn 5520  Oncon0 6159   Fn wfn 6319  ⟶wf 6320  –1-1→wf1 6321  –onto→wfo 6322  –1-1-onto→wf1o 6323  ‘cfv 6324   Isom wiso 6325  ωcom 7560  cardccrd 9348  ℵcale 9349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-oi 8958  df-har 9005  df-card 9352  df-aleph 9353

This theorem is referenced by:  alephiso2  40257