Theorem alephsuc2 9508

Description: An alternate representation of a successor aleph. The aleph function is the function obtained from the hartogs 9010 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 9371. (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		alephsucdom 9507	. . 3 ⊢ (𝐴 ∈ On → (𝑥 ≼ (ℵ‘𝐴) ↔ 𝑥 ≺ (ℵ‘suc 𝐴)))
2	1	rabbidv 3482	. 2 ⊢ (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘suc 𝐴)})
3		alephon 9497	. . . . . . 7 ⊢ (ℵ‘suc 𝐴) ∈ On
4	3	oneli 6300	. . . . . 6 ⊢ (𝑦 ∈ (ℵ‘suc 𝐴) → 𝑦 ∈ On)
5		alephcard 9498	. . . . . . . . 9 ⊢ (card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)
6		iscard 9406	. . . . . . . . 9 ⊢ ((card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴) ↔ ((ℵ‘suc 𝐴) ∈ On ∧ ∀𝑦 ∈ (ℵ‘suc 𝐴)𝑦 ≺ (ℵ‘suc 𝐴)))
7	5, 6	mpbi 232	. . . . . . . 8 ⊢ ((ℵ‘suc 𝐴) ∈ On ∧ ∀𝑦 ∈ (ℵ‘suc 𝐴)𝑦 ≺ (ℵ‘suc 𝐴))
8	7	simpri 488	. . . . . . 7 ⊢ ∀𝑦 ∈ (ℵ‘suc 𝐴)𝑦 ≺ (ℵ‘suc 𝐴)
9	8	rspec 3209	. . . . . 6 ⊢ (𝑦 ∈ (ℵ‘suc 𝐴) → 𝑦 ≺ (ℵ‘suc 𝐴))
10	4, 9	jca 514	. . . . 5 ⊢ (𝑦 ∈ (ℵ‘suc 𝐴) → (𝑦 ∈ On ∧ 𝑦 ≺ (ℵ‘suc 𝐴)))
11		sdomel 8666	. . . . . . 7 ⊢ ((𝑦 ∈ On ∧ (ℵ‘suc 𝐴) ∈ On) → (𝑦 ≺ (ℵ‘suc 𝐴) → 𝑦 ∈ (ℵ‘suc 𝐴)))
12	3, 11	mpan2 689	. . . . . 6 ⊢ (𝑦 ∈ On → (𝑦 ≺ (ℵ‘suc 𝐴) → 𝑦 ∈ (ℵ‘suc 𝐴)))
13	12	imp 409	. . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝑦 ≺ (ℵ‘suc 𝐴)) → 𝑦 ∈ (ℵ‘suc 𝐴))
14	10, 13	impbii 211	. . . 4 ⊢ (𝑦 ∈ (ℵ‘suc 𝐴) ↔ (𝑦 ∈ On ∧ 𝑦 ≺ (ℵ‘suc 𝐴)))
15		breq1 5071	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ≺ (ℵ‘suc 𝐴) ↔ 𝑦 ≺ (ℵ‘suc 𝐴)))
16	15	elrab 3682	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘suc 𝐴)} ↔ (𝑦 ∈ On ∧ 𝑦 ≺ (ℵ‘suc 𝐴)))
17	14, 16	bitr4i 280	. . 3 ⊢ (𝑦 ∈ (ℵ‘suc 𝐴) ↔ 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘suc 𝐴)})
18	17	eqriv 2820	. 2 ⊢ (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘suc 𝐴)}
19	2, 18	syl6reqr 2877	1 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)})

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  {crab 3144   class class class wbr 5068  Oncon0 6193  suc csuc 6195  ‘cfv 6357   ≼ cdom 8509   ≺ csdm 8510  cardccrd 9366  ℵcale 9367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-inf2 9106

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-se 5517  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-riota 7116  df-om 7583  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-oi 8976  df-har 9024  df-card 9370  df-aleph 9371

This theorem is referenced by:  alephsuc3  10004