Theorem alephsuc2 10103

Description: An alternate representation of a successor aleph. The aleph function is the function obtained from the hartogs 9567 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 9963. (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 10092	. . . . . . 7 ⊢ (ℵ‘suc 𝐴) ∈ On
2	1	oneli 6478	. . . . . 6 ⊢ (𝑦 ∈ (ℵ‘suc 𝐴) → 𝑦 ∈ On)
3		alephcard 10093	. . . . . . . . 9 ⊢ (card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)
4		iscard 9998	. . . . . . . . 9 ⊢ ((card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴) ↔ ((ℵ‘suc 𝐴) ∈ On ∧ ∀𝑦 ∈ (ℵ‘suc 𝐴)𝑦 ≺ (ℵ‘suc 𝐴)))
5	3, 4	mpbi 229	. . . . . . . 8 ⊢ ((ℵ‘suc 𝐴) ∈ On ∧ ∀𝑦 ∈ (ℵ‘suc 𝐴)𝑦 ≺ (ℵ‘suc 𝐴))
6	5	simpri 484	. . . . . . 7 ⊢ ∀𝑦 ∈ (ℵ‘suc 𝐴)𝑦 ≺ (ℵ‘suc 𝐴)
7	6	rspec 3238	. . . . . 6 ⊢ (𝑦 ∈ (ℵ‘suc 𝐴) → 𝑦 ≺ (ℵ‘suc 𝐴))
8	2, 7	jca 510	. . . . 5 ⊢ (𝑦 ∈ (ℵ‘suc 𝐴) → (𝑦 ∈ On ∧ 𝑦 ≺ (ℵ‘suc 𝐴)))
9		sdomel 9147	. . . . . . 7 ⊢ ((𝑦 ∈ On ∧ (ℵ‘suc 𝐴) ∈ On) → (𝑦 ≺ (ℵ‘suc 𝐴) → 𝑦 ∈ (ℵ‘suc 𝐴)))
10	1, 9	mpan2 689	. . . . . 6 ⊢ (𝑦 ∈ On → (𝑦 ≺ (ℵ‘suc 𝐴) → 𝑦 ∈ (ℵ‘suc 𝐴)))
11	10	imp 405	. . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝑦 ≺ (ℵ‘suc 𝐴)) → 𝑦 ∈ (ℵ‘suc 𝐴))
12	8, 11	impbii 208	. . . 4 ⊢ (𝑦 ∈ (ℵ‘suc 𝐴) ↔ (𝑦 ∈ On ∧ 𝑦 ≺ (ℵ‘suc 𝐴)))
13		breq1 5146	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ≺ (ℵ‘suc 𝐴) ↔ 𝑦 ≺ (ℵ‘suc 𝐴)))
14	13	elrab 3674	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘suc 𝐴)} ↔ (𝑦 ∈ On ∧ 𝑦 ≺ (ℵ‘suc 𝐴)))
15	12, 14	bitr4i 277	. . 3 ⊢ (𝑦 ∈ (ℵ‘suc 𝐴) ↔ 𝑦 ∈ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘suc 𝐴)})
16	15	eqriv 2722	. 2 ⊢ (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘suc 𝐴)}
17		alephsucdom 10102	. . 3 ⊢ (𝐴 ∈ On → (𝑥 ≼ (ℵ‘𝐴) ↔ 𝑥 ≺ (ℵ‘suc 𝐴)))
18	17	rabbidv 3427	. 2 ⊢ (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘suc 𝐴)})
19	16, 18	eqtr4id 2784	1 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)})

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3051  {crab 3419   class class class wbr 5143  Oncon0 6364  suc csuc 6366  ‘cfv 6543   ≼ cdom 8960   ≺ csdm 8961  cardccrd 9958  ℵcale 9959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-inf2 9664

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7372  df-ov 7419  df-om 7869  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-oi 9533  df-har 9580  df-card 9962  df-aleph 9963

This theorem is referenced by:  alephsuc3  10603