Theorem alephsuc3 10489

Description: An alternate representation of a successor aleph. Compare alephsuc 9976 and alephsuc2 9988. Equality can be obtained by taking the card of the right-hand side then using alephcard 9978 and carden 10459. (Contributed by NM, 23-Oct-2004.)

Assertion

Ref	Expression
alephsuc3	⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) ≈ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)})

Distinct variable group:   𝑥,𝐴

Proof of Theorem alephsuc3

Step	Hyp	Ref	Expression
1		alephsuc2 9988	. . . . 5 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)})
2		alephcard 9978	. . . . . . 7 ⊢ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
3		alephon 9977	. . . . . . . . 9 ⊢ (ℵ‘𝐴) ∈ On
4		onenon 9859	. . . . . . . . 9 ⊢ ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
5	3, 4	ax-mp 5	. . . . . . . 8 ⊢ (ℵ‘𝐴) ∈ dom card
6		cardval2 9901	. . . . . . . 8 ⊢ ((ℵ‘𝐴) ∈ dom card → (card‘(ℵ‘𝐴)) = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)})
7	5, 6	ax-mp 5	. . . . . . 7 ⊢ (card‘(ℵ‘𝐴)) = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}
8	2, 7	eqtr3i 2759	. . . . . 6 ⊢ (ℵ‘𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}
9	8	a1i 11	. . . . 5 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)})
10	1, 9	difeq12d 4077	. . . 4 ⊢ (𝐴 ∈ On → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) = ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}))
11		difrab 4268	. . . . 5 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}) = {𝑥 ∈ On ∣ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))}
12		bren2 8918	. . . . . 6 ⊢ (𝑥 ≈ (ℵ‘𝐴) ↔ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴)))
13	12	rabbii 3402	. . . . 5 ⊢ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} = {𝑥 ∈ On ∣ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))}
14	11, 13	eqtr4i 2760	. . . 4 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}) = {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)}
15	10, 14	eqtr2di 2786	. . 3 ⊢ (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} = ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)))
16		alephon 9977	. . . . 5 ⊢ (ℵ‘suc 𝐴) ∈ On
17		onenon 9859	. . . . 5 ⊢ ((ℵ‘suc 𝐴) ∈ On → (ℵ‘suc 𝐴) ∈ dom card)
18	16, 17	mp1i 13	. . . 4 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) ∈ dom card)
19		onsucb 7757	. . . . . 6 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
20		alephgeom 9990	. . . . . 6 ⊢ (suc 𝐴 ∈ On ↔ ω ⊆ (ℵ‘suc 𝐴))
21	19, 20	bitri 275	. . . . 5 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘suc 𝐴))
22		fvex 6845	. . . . . 6 ⊢ (ℵ‘suc 𝐴) ∈ V
23		ssdomg 8935	. . . . . 6 ⊢ ((ℵ‘suc 𝐴) ∈ V → (ω ⊆ (ℵ‘suc 𝐴) → ω ≼ (ℵ‘suc 𝐴)))
24	22, 23	ax-mp 5	. . . . 5 ⊢ (ω ⊆ (ℵ‘suc 𝐴) → ω ≼ (ℵ‘suc 𝐴))
25	21, 24	sylbi 217	. . . 4 ⊢ (𝐴 ∈ On → ω ≼ (ℵ‘suc 𝐴))
26		alephordilem1 9981	. . . 4 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
27		infdif 10116	. . . 4 ⊢ (((ℵ‘suc 𝐴) ∈ dom card ∧ ω ≼ (ℵ‘suc 𝐴) ∧ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)) → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) ≈ (ℵ‘suc 𝐴))
28	18, 25, 26, 27	syl3anc 1373	. . 3 ⊢ (𝐴 ∈ On → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) ≈ (ℵ‘suc 𝐴))
29	15, 28	eqbrtrd 5118	. 2 ⊢ (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} ≈ (ℵ‘suc 𝐴))
30	29	ensymd 8940	1 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) ≈ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {crab 3397  Vcvv 3438   ∖ cdif 3896   ⊆ wss 3899   class class class wbr 5096  dom cdm 5622  Oncon0 6315  suc csuc 6317  ‘cfv 6490  ωcom 7806   ≈ cen 8878   ≼ cdom 8879   ≺ csdm 8880  cardccrd 9845  ℵcale 9846

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-inf2 9548

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-oadd 8399  df-er 8633  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-oi 9413  df-har 9460  df-dju 9811  df-card 9849  df-aleph 9850

This theorem is referenced by: (None)