Theorem alephiso3 44560

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 44559	. 2 ⊢ ℵ Isom E , ≺ (On, {𝑥 ∈ ran card ∣ ω ⊆ 𝑥})
2		omelon 9647	. . . . . 6 ⊢ ω ∈ On
3		elrncard 44538	. . . . . . 7 ⊢ (𝑥 ∈ ran card ↔ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ¬ 𝑦 ≈ 𝑥))
4	3	simplbi 502	. . . . . 6 ⊢ (𝑥 ∈ ran card → 𝑥 ∈ On)
5		ontri1 6397	. . . . . 6 ⊢ ((ω ∈ On ∧ 𝑥 ∈ On) → (ω ⊆ 𝑥 ↔ ¬ 𝑥 ∈ ω))
6	2, 4, 5	sylancr 599	. . . . 5 ⊢ (𝑥 ∈ ran card → (ω ⊆ 𝑥 ↔ ¬ 𝑥 ∈ ω))
7	6	rabbiia 3416	. . . 4 ⊢ {𝑥 ∈ ran card ∣ ω ⊆ 𝑥} = {𝑥 ∈ ran card ∣ ¬ 𝑥 ∈ ω}
8		dfdif2 4048	. . . 4 ⊢ (ran card ∖ ω) = {𝑥 ∈ ran card ∣ ¬ 𝑥 ∈ ω}
9	7, 8	eqtr4i 2786	. . 3 ⊢ {𝑥 ∈ ran card ∣ ω ⊆ 𝑥} = (ran card ∖ ω)
10		isoeq5 7329	. . 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 233	1 ⊢ ℵ Isom E , ≺ (On, (ran card ∖ ω))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412   ⊆ wss 3895   ∖ cdif 4045   class class class wbr 5102   E cep 5549  ran crn 5651  Oncon0 6362   Isom wiso 6539  ωcom 7877   ≈ cen 8970   ≺ csdm 8972  cardccrd 10016  ℵcale 10017

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9642

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-oi 9504  df-har 9551  df-card 10020  df-aleph 10021

This theorem is used by: (None)