Theorem alephiso2 42611

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 10095	. 2 ⊢ ℵ Isom E , E (On, {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)})
2		iscard4 42586	. . . . . . . 8 ⊢ ((card‘𝑥) = 𝑥 ↔ 𝑥 ∈ ran card)
3	2	anbi1ci 624	. . . . . . 7 ⊢ ((ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥) ↔ (𝑥 ∈ ran card ∧ ω ⊆ 𝑥))
4	3	abbii 2800	. . . . . 6 ⊢ {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} = {𝑥 ∣ (𝑥 ∈ ran card ∧ ω ⊆ 𝑥)}
5		df-rab 3431	. . . . . 6 ⊢ {𝑥 ∈ ran card ∣ ω ⊆ 𝑥} = {𝑥 ∣ (𝑥 ∈ ran card ∧ ω ⊆ 𝑥)}
6	4, 5	eqtr4i 2761	. . . . 5 ⊢ {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} = {𝑥 ∈ ran card ∣ ω ⊆ 𝑥}
7		f1oeq3 6822	. . . . 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 10066	. . . . . . . . 9 ⊢ (ℵ‘𝑧) ∈ On
10		epelg 5580	. . . . . . . . 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 10073	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 ∈ 𝑧 ↔ (ℵ‘𝑦) ∈ (ℵ‘𝑧)))
13		alephord 10072	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 ∈ 𝑧 ↔ (ℵ‘𝑦) ≺ (ℵ‘𝑧)))
14	11, 12, 13	3bitr2d 306	. . . . . . 7 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → ((ℵ‘𝑦) E (ℵ‘𝑧) ↔ (ℵ‘𝑦) ≺ (ℵ‘𝑧)))
15	14	bibi2d 341	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → ((𝑦 E 𝑧 ↔ (ℵ‘𝑦) E (ℵ‘𝑧)) ↔ (𝑦 E 𝑧 ↔ (ℵ‘𝑦) ≺ (ℵ‘𝑧))))
16	15	ralbidva 3173	. . . . 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 3089	. . . 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 625	. . 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 6551	. . 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 6551	. . 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 302	. 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 229	1 ⊢ ℵ Isom E , ≺ (On, {𝑥 ∈ ran card ∣ ω ⊆ 𝑥})

Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  {cab 2707  ∀wral 3059  {crab 3430   ⊆ wss 3947   class class class wbr 5147   E cep 5578  ran crn 5676  Oncon0 6363  –1-1-onto→wf1o 6541  ‘cfv 6542   Isom wiso 6543  ωcom 7857   ≺ csdm 8940  cardccrd 9932  ℵcale 9933

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-oi 9507  df-har 9554  df-card 9936  df-aleph 9937

This theorem is referenced by:  alephiso3  42612