Theorem alephsuc2 10060

Description: An alternate representation of a successor aleph. The aleph function is the function obtained from the hartogs 9502 function by transfinite recursion, starting from ω. Using this theorem we could define the aleph function with {𝑧 ∈ On ∣ 𝑧 ≼ 𝑥} in place of ∩ {𝑧 ∈ On ∣ 𝑥 ≺ 𝑧} in df-aleph 9922. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)

Assertion

Ref	Expression
alephsuc2	⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)})

Distinct variable group:   𝑥,𝐴

Proof of Theorem alephsuc2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		alephon 10049	. . . . . . 7 ⊢ (ℵ‘suc 𝐴) ∈ On
2	1	oneli 6473	. . . . . 6 ⊢ (𝑦 ∈ (ℵ‘suc 𝐴) → 𝑦 ∈ On)
3		alephcard 10050	. . . . . . . . 9 ⊢ (card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)
4		iscard 9957	. . . . . . . . 9 ⊢ ((card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴) ↔ ((ℵ‘suc 𝐴) ∈ On ∧ ∀𝑦 ∈ (ℵ‘suc 𝐴)𝑦 ≺ (ℵ‘suc 𝐴)))
5	3, 4	mpbi 233	. . . . . . . 8 ⊢ ((ℵ‘suc 𝐴) ∈ On ∧ ∀𝑦 ∈ (ℵ‘suc 𝐴)𝑦 ≺ (ℵ‘suc 𝐴))
6	5	simpri 490	. . . . . . 7 ⊢ ∀𝑦 ∈ (ℵ‘suc 𝐴)𝑦 ≺ (ℵ‘suc 𝐴)
7	6	rspec 3262	. . . . . 6 ⊢ (𝑦 ∈ (ℵ‘suc 𝐴) → 𝑦 ≺ (ℵ‘suc 𝐴))
8	2, 7	jca 520	. . . . 5 ⊢ (𝑦 ∈ (ℵ‘suc 𝐴) → (𝑦 ∈ On ∧ 𝑦 ≺ (ℵ‘suc 𝐴)))
9		sdomel 9108	. . . . . . 7 ⊢ ((𝑦 ∈ On ∧ (ℵ‘suc 𝐴) ∈ On) → (𝑦 ≺ (ℵ‘suc 𝐴) → 𝑦 ∈ (ℵ‘suc 𝐴)))
10	1, 9	mpan2 703	. . . . . 6 ⊢ (𝑦 ∈ On → (𝑦 ≺ (ℵ‘suc 𝐴) → 𝑦 ∈ (ℵ‘suc 𝐴)))
11	10	imp 411	. . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝑦 ≺ (ℵ‘suc 𝐴)) → 𝑦 ∈ (ℵ‘suc 𝐴))
12	8, 11	impbii 212	. . . 4 ⊢ (𝑦 ∈ (ℵ‘suc 𝐴) ↔ (𝑦 ∈ On ∧ 𝑦 ≺ (ℵ‘suc 𝐴)))
13		breq1 5113	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ≺ (ℵ‘suc 𝐴) ↔ 𝑦 ≺ (ℵ‘suc 𝐴)))
14	13	elrab 3659	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘suc 𝐴)} ↔ (𝑦 ∈ On ∧ 𝑦 ≺ (ℵ‘suc 𝐴)))
15	12, 14	bitr4i 281	. . 3 ⊢ (𝑦 ∈ (ℵ‘suc 𝐴) ↔ 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘suc 𝐴)})
16	15	eqriv 2766	. 2 ⊢ (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘suc 𝐴)}
17		alephsucdom 10059	. . 3 ⊢ (𝐴 ∈ On → (𝑥 ≼ (ℵ‘𝐴) ↔ 𝑥 ≺ (ℵ‘suc 𝐴)))
18	17	rabbidv 3430	. 2 ⊢ (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘suc 𝐴)})
19	16, 18	eqtr4id 2823	1 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)})

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  {crab 3423   class class class wbr 5110  Oncon0 6357  suc csuc 6359  ‘cfv 6533   ≼ cdom 8937   ≺ 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:  alephsuc3  10561