Theorem alephiso2 39966

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 9524	. 2 ⊢ ℵ Isom E , E (On, {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)})
2		iscard4 39949	. . . . . . . 8 ⊢ ((card‘𝑥) = 𝑥 ↔ 𝑥 ∈ ran card)
3	2	anbi1ci 627	. . . . . . 7 ⊢ ((ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥) ↔ (𝑥 ∈ ran card ∧ ω ⊆ 𝑥))
4	3	abbii 2886	. . . . . 6 ⊢ {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} = {𝑥 ∣ (𝑥 ∈ ran card ∧ ω ⊆ 𝑥)}
5		df-rab 3147	. . . . . 6 ⊢ {𝑥 ∈ ran card ∣ ω ⊆ 𝑥} = {𝑥 ∣ (𝑥 ∈ ran card ∧ ω ⊆ 𝑥)}
6	4, 5	eqtr4i 2847	. . . . 5 ⊢ {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} = {𝑥 ∈ ran card ∣ ω ⊆ 𝑥}
7		f1oeq3 6606	. . . . 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 9495	. . . . . . . . 9 ⊢ (ℵ‘𝑧) ∈ On
10		epelg 5466	. . . . . . . . 9 ⊢ ((ℵ‘𝑧) ∈ On → ((ℵ‘𝑦) E (ℵ‘𝑧) ↔ (ℵ‘𝑦) ∈ (ℵ‘𝑧)))
11	9, 10	mp1i 13	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → ((ℵ‘𝑦) E (ℵ‘𝑧) ↔ (ℵ‘𝑦) ∈ (ℵ‘𝑧)))
12		alephord2 9502	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 ∈ 𝑧 ↔ (ℵ‘𝑦) ∈ (ℵ‘𝑧)))
13		alephord 9501	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 ∈ 𝑧 ↔ (ℵ‘𝑦) ≺ (ℵ‘𝑧)))
14	11, 12, 13	3bitr2d 309	. . . . . . 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 3196	. . . . 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 3164	. . . 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 628	. . 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 6364	. . 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 6364	. . 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 305	. 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 232	1 ⊢ ℵ Isom E , ≺ (On, {𝑥 ∈ ran card ∣ ω ⊆ 𝑥})

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {cab 2799  ∀wral 3138  {crab 3142   ⊆ wss 3936   class class class wbr 5066   E cep 5464  ran crn 5556  Oncon0 6191  –1-1-onto→wf1o 6354  ‘cfv 6355   Isom wiso 6356  ωcom 7580   ≺ csdm 8508  cardccrd 9364  ℵcale 9365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-inf2 9104

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-om 7581  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-oi 8974  df-har 9022  df-card 9368  df-aleph 9369

This theorem is referenced by:  alephiso3  39967