Theorem alephsuc2 10099

Description: An alternate representation of a successor aleph. The aleph function is the function obtained from the hartogs 9563 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 9959. (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 10088	. . . . . . 7 ⊢ (ℵ‘suc 𝐴) ∈ On
2	1	oneli 6473	. . . . . 6 ⊢ (𝑦 ∈ (ℵ‘suc 𝐴) → 𝑦 ∈ On)
3		alephcard 10089	. . . . . . . . 9 ⊢ (card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)
4		iscard 9994	. . . . . . . . 9 ⊢ ((card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴) ↔ ((ℵ‘suc 𝐴) ∈ On ∧ ∀𝑦 ∈ (ℵ‘suc 𝐴)𝑦 ≺ (ℵ‘suc 𝐴)))
5	3, 4	mpbi 230	. . . . . . . 8 ⊢ ((ℵ‘suc 𝐴) ∈ On ∧ ∀𝑦 ∈ (ℵ‘suc 𝐴)𝑦 ≺ (ℵ‘suc 𝐴))
6	5	simpri 485	. . . . . . 7 ⊢ ∀𝑦 ∈ (ℵ‘suc 𝐴)𝑦 ≺ (ℵ‘suc 𝐴)
7	6	rspec 3237	. . . . . 6 ⊢ (𝑦 ∈ (ℵ‘suc 𝐴) → 𝑦 ≺ (ℵ‘suc 𝐴))
8	2, 7	jca 511	. . . . 5 ⊢ (𝑦 ∈ (ℵ‘suc 𝐴) → (𝑦 ∈ On ∧ 𝑦 ≺ (ℵ‘suc 𝐴)))
9		sdomel 9143	. . . . . . 7 ⊢ ((𝑦 ∈ On ∧ (ℵ‘suc 𝐴) ∈ On) → (𝑦 ≺ (ℵ‘suc 𝐴) → 𝑦 ∈ (ℵ‘suc 𝐴)))
10	1, 9	mpan2 691	. . . . . 6 ⊢ (𝑦 ∈ On → (𝑦 ≺ (ℵ‘suc 𝐴) → 𝑦 ∈ (ℵ‘suc 𝐴)))
11	10	imp 406	. . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝑦 ≺ (ℵ‘suc 𝐴)) → 𝑦 ∈ (ℵ‘suc 𝐴))
12	8, 11	impbii 209	. . . 4 ⊢ (𝑦 ∈ (ℵ‘suc 𝐴) ↔ (𝑦 ∈ On ∧ 𝑦 ≺ (ℵ‘suc 𝐴)))
13		breq1 5127	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ≺ (ℵ‘suc 𝐴) ↔ 𝑦 ≺ (ℵ‘suc 𝐴)))
14	13	elrab 3676	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘suc 𝐴)} ↔ (𝑦 ∈ On ∧ 𝑦 ≺ (ℵ‘suc 𝐴)))
15	12, 14	bitr4i 278	. . 3 ⊢ (𝑦 ∈ (ℵ‘suc 𝐴) ↔ 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘suc 𝐴)})
16	15	eqriv 2733	. 2 ⊢ (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘suc 𝐴)}
17		alephsucdom 10098	. . 3 ⊢ (𝐴 ∈ On → (𝑥 ≼ (ℵ‘𝐴) ↔ 𝑥 ≺ (ℵ‘suc 𝐴)))
18	17	rabbidv 3428	. 2 ⊢ (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘suc 𝐴)})
19	16, 18	eqtr4id 2790	1 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3052  {crab 3420   class class class wbr 5124  Oncon0 6357  suc csuc 6359  ‘cfv 6536   ≼ cdom 8962   ≺ csdm 8963  cardccrd 9954  ℵcale 9955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-inf2 9660

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  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 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-om 7867  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-oi 9529  df-har 9576  df-card 9958  df-aleph 9959

This theorem is referenced by:  alephsuc3  10599