Theorem alephiso 10148

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 10115	. . . . . 6 ⊢ ℵ Fn On
2		isinfcard 10142	. . . . . . . 8 ⊢ ((ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥) ↔ 𝑥 ∈ ran ℵ)
3	2	bicomi 227	. . . . . . 7 ⊢ (𝑥 ∈ ran ℵ ↔ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥))
4	3	eqabi 2895	. . . . . 6 ⊢ ran ℵ = {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)}
5		df-fo 6533	. . . . . 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 724	. . . . 5 ⊢ ℵ:On–onto→{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)}
7		fof 6784	. . . . 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 10134	. . . . . 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 3202	. . . 4 ⊢ ∀𝑦 ∈ On ∀𝑧 ∈ On ((ℵ‘𝑦) = (ℵ‘𝑧) → 𝑦 = 𝑧)
12		dff13 7246	. . . 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 724	. . 3 ⊢ ℵ:On–1-1→{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)}
14		df-f1o 6534	. . 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 724	. 2 ⊢ ℵ:On–1-1-onto→{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)}
16		alephord2 10126	. . . 4 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 ∈ 𝑧 ↔ (ℵ‘𝑦) ∈ (ℵ‘𝑧)))
17		epel 5550	. . . 4 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧)
18		fvex 6886	. . . . 5 ⊢ (ℵ‘𝑧) ∈ V
19	18	epeli 5549	. . . 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 3202	. 2 ⊢ ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑦 E 𝑧 ↔ (ℵ‘𝑦) E (ℵ‘𝑧))
22		df-isom 6536	. 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 724	1 ⊢ ℵ Isom E , E (On, {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2738  ∀wral 3076   ⊆ wss 3898   class class class wbr 5102   E cep 5546  ran crn 5648  Oncon0 6351   Fn wfn 6522  ⟶wf 6523  –1-1→wf1 6524  –onto→wfo 6525  –1-1-onto→wf1o 6526  ‘cfv 6527   Isom wiso 6528  ωcom 7860  cardccrd 9987  ℵcale 9988

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-inf2 9620

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-isom 6536  df-riota 7365  df-ov 7411  df-om 7861  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-er 8695  df-en 8952  df-dom 8953  df-sdom 8954  df-fin 8955  df-oi 9482  df-har 9529  df-card 9991  df-aleph 9992

This theorem is used by:  alephiso2  44502