Theorem alephiso 9992

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 9959	. . . . . 6 ⊢ ℵ Fn On
2		isinfcard 9986	. . . . . . . 8 ⊢ ((ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥) ↔ 𝑥 ∈ ran ℵ)
3	2	bicomi 224	. . . . . . 7 ⊢ (𝑥 ∈ ran ℵ ↔ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥))
4	3	eqabi 2863	. . . . . 6 ⊢ ran ℵ = {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)}
5		df-fo 6488	. . . . . 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 711	. . . . 5 ⊢ ℵ:On–onto→{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)}
7		fof 6736	. . . . 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 9978	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → ((ℵ‘𝑦) = (ℵ‘𝑧) ↔ 𝑦 = 𝑧))
10	9	biimpd 229	. . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → ((ℵ‘𝑦) = (ℵ‘𝑧) → 𝑦 = 𝑧))
11	10	rgen2 3169	. . . 4 ⊢ ∀𝑦 ∈ On ∀𝑧 ∈ On ((ℵ‘𝑦) = (ℵ‘𝑧) → 𝑦 = 𝑧)
12		dff13 7191	. . . 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 711	. . 3 ⊢ ℵ:On–1-1→{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)}
14		df-f1o 6489	. . 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 711	. 2 ⊢ ℵ:On–1-1-onto→{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)}
16		alephord2 9970	. . . 4 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 ∈ 𝑧 ↔ (ℵ‘𝑦) ∈ (ℵ‘𝑧)))
17		epel 5522	. . . 4 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧)
18		fvex 6835	. . . . 5 ⊢ (ℵ‘𝑧) ∈ V
19	18	epeli 5521	. . . 4 ⊢ ((ℵ‘𝑦) E (ℵ‘𝑧) ↔ (ℵ‘𝑦) ∈ (ℵ‘𝑧))
20	16, 17, 19	3bitr4g 314	. . 3 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 E 𝑧 ↔ (ℵ‘𝑦) E (ℵ‘𝑧)))
21	20	rgen2 3169	. 2 ⊢ ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑦 E 𝑧 ↔ (ℵ‘𝑦) E (ℵ‘𝑧))
22		df-isom 6491	. 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 711	1 ⊢ ℵ Isom E , E (On, {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044   ⊆ wss 3903   class class class wbr 5092   E cep 5518  ran crn 5620  Oncon0 6307   Fn wfn 6477  ⟶wf 6478  –1-1→wf1 6479  –onto→wfo 6480  –1-1-onto→wf1o 6481  ‘cfv 6482   Isom wiso 6483  ωcom 7799  cardccrd 9831  ℵcale 9832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-inf2 9537

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-oi 9402  df-har 9449  df-card 9835  df-aleph 9836

This theorem is referenced by:  alephiso2  43541