Theorem alephsuc3 10540

Description: An alternate representation of a successor aleph. Compare alephsuc 10026 and alephsuc2 10038. Equality can be obtained by taking the card of the right-hand side then using alephcard 10028 and carden 10510. (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 10038	. . . . 5 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)})
2		alephcard 10028	. . . . . . 7 ⊢ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
3		alephon 10027	. . . . . . . . 9 ⊢ (ℵ‘𝐴) ∈ On
4		onenon 9909	. . . . . . . . 9 ⊢ ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
5	3, 4	ax-mp 5	. . . . . . . 8 ⊢ (ℵ‘𝐴) ∈ dom card
6		cardval2 9951	. . . . . . . 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 2789	. . . . . 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 4083	. . . 4 ⊢ (𝐴 ∈ On → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) = ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}))
11		difrab 4272	. . . . 5 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}) = {𝑥 ∈ On ∣ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))}
12		bren2 8966	. . . . . 6 ⊢ (𝑥 ≈ (ℵ‘𝐴) ↔ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴)))
13	12	rabbii 3421	. . . . 5 ⊢ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} = {𝑥 ∈ On ∣ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))}
14	11, 13	eqtr4i 2790	. . . 4 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}) = {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)}
15	10, 14	eqtr2di 2816	. . 3 ⊢ (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} = ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)))
16		alephon 10027	. . . . 5 ⊢ (ℵ‘suc 𝐴) ∈ On
17		onenon 9909	. . . . 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 7799	. . . . . 6 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
20		alephgeom 10040	. . . . . 6 ⊢ (suc 𝐴 ∈ On ↔ ω ⊆ (ℵ‘suc 𝐴))
21	19, 20	bitri 277	. . . . 5 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘suc 𝐴))
22		fvex 6882	. . . . . 6 ⊢ (ℵ‘suc 𝐴) ∈ V
23		ssdomg 8983	. . . . . 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 219	. . . 4 ⊢ (𝐴 ∈ On → ω ≼ (ℵ‘suc 𝐴))
26		alephordilem1 10031	. . . 4 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
27		infdif 10166	. . . 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 1392	. . 3 ⊢ (𝐴 ∈ On → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) ≈ (ℵ‘suc 𝐴))
29	15, 28	eqbrtrd 5124	. 2 ⊢ (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} ≈ (ℵ‘suc 𝐴))
30	29	ensymd 8988	1 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) ≈ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144  {crab 3416  Vcvv 3456   ∖ cdif 3903   ⊆ wss 3906   class class class wbr 5102  dom cdm 5649  Oncon0 6348  suc csuc 6350  ‘cfv 6523  ωcom 7848   ≈ cen 8926   ≼ cdom 8927   ≺ csdm 8928  cardccrd 9895  ℵcale 9896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-inf2 9598

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-se 5603  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-isom 6532  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-2o 8440  df-oadd 8443  df-er 8680  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-oi 9460  df-har 9507  df-dju 9861  df-card 9899  df-aleph 9900

This theorem is referenced by: (None)