Theorem alephiso2 44398

Description: ℵ is a strictly order-preserving mapping of On onto the class of all infinite cardinal numbers. (Contributed by RP, 18-Nov-2023.)

Assertion

Ref	Expression
alephiso2	⊢ ℵ Isom E , ≺ (On, {𝑥 ∈ ran card ∣ ω ⊆ 𝑥})

Proof of Theorem alephiso2

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		alephiso 10101	. 2 ⊢ ℵ Isom E , E (On, {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)})
2		iscard4 44373	. . . . . . . 8 ⊢ ((card‘𝑥) = 𝑥 ↔ 𝑥 ∈ ran card)
3	2	anbi1ci 638	. . . . . . 7 ⊢ ((ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥) ↔ (𝑥 ∈ ran card ∧ ω ⊆ 𝑥))
4	3	abbii 2827	. . . . . 6 ⊢ {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} = {𝑥 ∣ (𝑥 ∈ ran card ∧ ω ⊆ 𝑥)}
5		df-rab 3413	. . . . . 6 ⊢ {𝑥 ∈ ran card ∣ ω ⊆ 𝑥} = {𝑥 ∣ (𝑥 ∈ ran card ∧ ω ⊆ 𝑥)}
6	4, 5	eqtr4i 2786	. . . . 5 ⊢ {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} = {𝑥 ∈ ran card ∣ ω ⊆ 𝑥}
7		f1oeq3 6807	. . . . 5 ⊢ ({𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} = {𝑥 ∈ ran card ∣ ω ⊆ 𝑥} → (ℵ:On–1-1-onto→{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} ↔ ℵ:On–1-1-onto→{𝑥 ∈ ran card ∣ ω ⊆ 𝑥}))
8	6, 7	ax-mp 5	. . . 4 ⊢ (ℵ:On–1-1-onto→{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} ↔ ℵ:On–1-1-onto→{𝑥 ∈ ran card ∣ ω ⊆ 𝑥})
9		alephon 10072	. . . . . . . . 9 ⊢ (ℵ‘𝑧) ∈ On
10		epelg 5556	. . . . . . . . 9 ⊢ ((ℵ‘𝑧) ∈ On → ((ℵ‘𝑦) E (ℵ‘𝑧) ↔ (ℵ‘𝑦) ∈ (ℵ‘𝑧)))
11	9, 10	mp1i 14	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → ((ℵ‘𝑦) E (ℵ‘𝑧) ↔ (ℵ‘𝑦) ∈ (ℵ‘𝑧)))
12		alephord2 10079	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 ∈ 𝑧 ↔ (ℵ‘𝑦) ∈ (ℵ‘𝑧)))
13		alephord 10078	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 ∈ 𝑧 ↔ (ℵ‘𝑦) ≺ (ℵ‘𝑧)))
14	11, 12, 13	3bitr2d 310	. . . . . . 7 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → ((ℵ‘𝑦) E (ℵ‘𝑧) ↔ (ℵ‘𝑦) ≺ (ℵ‘𝑧)))
15	14	bibi2d 345	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → ((𝑦 E 𝑧 ↔ (ℵ‘𝑦) E (ℵ‘𝑧)) ↔ (𝑦 E 𝑧 ↔ (ℵ‘𝑦) ≺ (ℵ‘𝑧))))
16	15	ralbidva 3183	. . . . 5 ⊢ (𝑦 ∈ On → (∀𝑧 ∈ On (𝑦 E 𝑧 ↔ (ℵ‘𝑦) E (ℵ‘𝑧)) ↔ ∀𝑧 ∈ On (𝑦 E 𝑧 ↔ (ℵ‘𝑦) ≺ (ℵ‘𝑧))))
17	16	ralbiia 3106	. . . 4 ⊢ (∀𝑦 ∈ On ∀𝑧 ∈ On (𝑦 E 𝑧 ↔ (ℵ‘𝑦) E (ℵ‘𝑧)) ↔ ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑦 E 𝑧 ↔ (ℵ‘𝑦) ≺ (ℵ‘𝑧)))
18	8, 17	anbi12i 640	. . 3 ⊢ ((ℵ:On–1-1-onto→{𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} ∧ ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑦 E 𝑧 ↔ (ℵ‘𝑦) E (ℵ‘𝑧))) ↔ (ℵ:On–1-1-onto→{𝑥 ∈ ran card ∣ ω ⊆ 𝑥} ∧ ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑦 E 𝑧 ↔ (ℵ‘𝑦) ≺ (ℵ‘𝑧))))
19		df-isom 6542	. . 3 ⊢ (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))))
20		df-isom 6542	. . 3 ⊢ (ℵ Isom E , ≺ (On, {𝑥 ∈ ran card ∣ ω ⊆ 𝑥}) ↔ (ℵ:On–1-1-onto→{𝑥 ∈ ran card ∣ ω ⊆ 𝑥} ∧ ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑦 E 𝑧 ↔ (ℵ‘𝑦) ≺ (ℵ‘𝑧))))
21	18, 19, 20	3bitr4i 306	. 2 ⊢ (ℵ Isom E , E (On, {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)}) ↔ ℵ Isom E , ≺ (On, {𝑥 ∈ ran card ∣ ω ⊆ 𝑥}))
22	1, 21	mpbi 233	1 ⊢ ℵ Isom E , ≺ (On, {𝑥 ∈ ran card ∣ ω ⊆ 𝑥})

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2738  ∀wral 3076  {crab 3412   ⊆ wss 3899   class class class wbr 5103   E cep 5554  ran crn 5656  Oncon0 6357  –1-1-onto→wf1o 6532  ‘cfv 6533   Isom wiso 6534  ωcom 7862   ≺ csdm 8951  cardccrd 9940  ℵcale 9941

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 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  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 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7370  df-ov 7416  df-om 7863  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-oi 9482  df-har 9529  df-card 9944  df-aleph 9945

This theorem is used by:  alephiso3  44399