Theorem alephiso3 42295

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
alephiso3	⊢ ℵ Isom E , ≺ (On, (ran card ∖ ω))

Proof of Theorem alephiso3

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

Step	Hyp	Ref	Expression
1		alephiso2 42294	. 2 ⊢ ℵ Isom E , ≺ (On, {𝑥 ∈ ran card ∣ ω ⊆ 𝑥})
2		omelon 9637	. . . . . 6 ⊢ ω ∈ On
3		elrncard 42273	. . . . . . 7 ⊢ (𝑥 ∈ ran card ↔ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ¬ 𝑦 ≈ 𝑥))
4	3	simplbi 498	. . . . . 6 ⊢ (𝑥 ∈ ran card → 𝑥 ∈ On)
5		ontri1 6395	. . . . . 6 ⊢ ((ω ∈ On ∧ 𝑥 ∈ On) → (ω ⊆ 𝑥 ↔ ¬ 𝑥 ∈ ω))
6	2, 4, 5	sylancr 587	. . . . 5 ⊢ (𝑥 ∈ ran card → (ω ⊆ 𝑥 ↔ ¬ 𝑥 ∈ ω))
7	6	rabbiia 3436	. . . 4 ⊢ {𝑥 ∈ ran card ∣ ω ⊆ 𝑥} = {𝑥 ∈ ran card ∣ ¬ 𝑥 ∈ ω}
8		dfdif2 3956	. . . 4 ⊢ (ran card ∖ ω) = {𝑥 ∈ ran card ∣ ¬ 𝑥 ∈ ω}
9	7, 8	eqtr4i 2763	. . 3 ⊢ {𝑥 ∈ ran card ∣ ω ⊆ 𝑥} = (ran card ∖ ω)
10		isoeq5 7314	. . 3 ⊢ ({𝑥 ∈ ran card ∣ ω ⊆ 𝑥} = (ran card ∖ ω) → (ℵ Isom E , ≺ (On, {𝑥 ∈ ran card ∣ ω ⊆ 𝑥}) ↔ ℵ Isom E , ≺ (On, (ran card ∖ ω))))
11	9, 10	ax-mp 5	. 2 ⊢ (ℵ Isom E , ≺ (On, {𝑥 ∈ ran card ∣ ω ⊆ 𝑥}) ↔ ℵ Isom E , ≺ (On, (ran card ∖ ω)))
12	1, 11	mpbi 229	1 ⊢ ℵ Isom E , ≺ (On, (ran card ∖ ω))

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1541   ∈ wcel 2106  ∀wral 3061  {crab 3432   ∖ cdif 3944   ⊆ wss 3947   class class class wbr 5147   E cep 5578  ran crn 5676  Oncon0 6361   Isom wiso 6541  ωcom 7851   ≈ cen 8932   ≺ csdm 8934  cardccrd 9926  ℵcale 9927

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  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 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-om 7852  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-oi 9501  df-har 9548  df-card 9930  df-aleph 9931

This theorem is referenced by: (None)