Theorem alephiso2 44169

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 10078	. 2 ⊢ ℵ Isom E , E (On, {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)})
2		iscard4 44144	. . . . . . . 8 ⊢ ((card‘𝑥) = 𝑥 ↔ 𝑥 ∈ ran card)
3	2	anbi1ci 637	. . . . . . 7 ⊢ ((ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥) ↔ (𝑥 ∈ ran card ∧ ω ⊆ 𝑥))
4	3	abbii 2836	. . . . . 6 ⊢ {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} = {𝑥 ∣ (𝑥 ∈ ran card ∧ ω ⊆ 𝑥)}
5		df-rab 3424	. . . . . 6 ⊢ {𝑥 ∈ ran card ∣ ω ⊆ 𝑥} = {𝑥 ∣ (𝑥 ∈ ran card ∧ ω ⊆ 𝑥)}
6	4, 5	eqtr4i 2795	. . . . 5 ⊢ {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} = {𝑥 ∈ ran card ∣ ω ⊆ 𝑥}
7		f1oeq3 6808	. . . . 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 10049	. . . . . . . . 9 ⊢ (ℵ‘𝑧) ∈ On
10		epelg 5560	. . . . . . . . 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 10056	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 ∈ 𝑧 ↔ (ℵ‘𝑦) ∈ (ℵ‘𝑧)))
13		alephord 10055	. . . . . . . 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 3192	. . . . 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 3115	. . . 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 639	. . 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})

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {cab 2747  ∀wral 3085  {crab 3423   ⊆ wss 3913   class class class wbr 5110   E cep 5558  ran crn 5660  Oncon0 6357  –1-1-onto→wf1o 6532  ‘cfv 6533   Isom wiso 6534  ωcom 7858   ≺ csdm 8938  cardccrd 9917  ℵcale 9918

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-inf2 9606

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  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 7365  df-ov 7411  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-oi 9468  df-har 9515  df-card 9921  df-aleph 9922

This theorem is referenced by:  alephiso3  44170