Theorem alephiso2 43835

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 10012	. 2 ⊢ ℵ Isom E , E (On, {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)})
2		iscard4 43810	. . . . . . . 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 2804	. . . . . 6 ⊢ {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} = {𝑥 ∣ (𝑥 ∈ ran card ∧ ω ⊆ 𝑥)}
5		df-rab 3401	. . . . . 6 ⊢ {𝑥 ∈ ran card ∣ ω ⊆ 𝑥} = {𝑥 ∣ (𝑥 ∈ ran card ∧ ω ⊆ 𝑥)}
6	4, 5	eqtr4i 2763	. . . . 5 ⊢ {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)} = {𝑥 ∈ ran card ∣ ω ⊆ 𝑥}
7		f1oeq3 6765	. . . . 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 9983	. . . . . . . . 9 ⊢ (ℵ‘𝑧) ∈ On
10		epelg 5526	. . . . . . . . 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 9990	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 ∈ 𝑧 ↔ (ℵ‘𝑦) ∈ (ℵ‘𝑧)))
13		alephord 9989	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 ∈ 𝑧 ↔ (ℵ‘𝑦) ≺ (ℵ‘𝑧)))
14	11, 12, 13	3bitr2d 307	. . . . . . 7 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → ((ℵ‘𝑦) E (ℵ‘𝑧) ↔ (ℵ‘𝑦) ≺ (ℵ‘𝑧)))
15	14	bibi2d 342	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → ((𝑦 E 𝑧 ↔ (ℵ‘𝑦) E (ℵ‘𝑧)) ↔ (𝑦 E 𝑧 ↔ (ℵ‘𝑦) ≺ (ℵ‘𝑧))))
16	15	ralbidva 3158	. . . . 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 3081	. . . 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 629	. . 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 6502	. . 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 6502	. . 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 303	. 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 230	1 ⊢ ℵ Isom E , ≺ (On, {𝑥 ∈ ran card ∣ ω ⊆ 𝑥})

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  {crab 3400   ⊆ wss 3902   class class class wbr 5099   E cep 5524  ran crn 5626  Oncon0 6318  –1-1-onto→wf1o 6492  ‘cfv 6493   Isom wiso 6494  ωcom 7810   ≺ csdm 8886  cardccrd 9851  ℵcale 9852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-inf2 9554

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7317  df-ov 7363  df-om 7811  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-oi 9419  df-har 9466  df-card 9855  df-aleph 9856

This theorem is referenced by:  alephiso3  43836