Theorem alephsuc3 10636

Description: An alternate representation of a successor aleph. Compare alephsuc 10118 and alephsuc2 10130. Equality can be obtained by taking the card of the right-hand side then using alephcard 10120 and carden 10606. (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 10130	. . . . 5 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)})
2		alephcard 10120	. . . . . . 7 ⊢ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
3		alephon 10119	. . . . . . . . 9 ⊢ (ℵ‘𝐴) ∈ On
4		onenon 10001	. . . . . . . . 9 ⊢ ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
5	3, 4	ax-mp 5	. . . . . . . 8 ⊢ (ℵ‘𝐴) ∈ dom card
6		cardval2 10043	. . . . . . . 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 2785	. . . . . 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 4074	. . . 4 ⊢ (𝐴 ∈ On → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) = ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}))
11		difrab 4263	. . . . 5 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}) = {𝑥 ∈ On ∣ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))}
12		bren2 8988	. . . . . 6 ⊢ (𝑥 ≈ (ℵ‘𝐴) ↔ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴)))
13	12	rabbii 3417	. . . . 5 ⊢ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} = {𝑥 ∈ On ∣ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))}
14	11, 13	eqtr4i 2786	. . . 4 ⊢ ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}) = {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)}
15	10, 14	eqtr2di 2812	. . 3 ⊢ (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} = ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)))
16		alephon 10119	. . . . 5 ⊢ (ℵ‘suc 𝐴) ∈ On
17		onenon 10001	. . . . 5 ⊢ ((ℵ‘suc 𝐴) ∈ On → (ℵ‘suc 𝐴) ∈ dom card)
18	16, 17	mp1i 14	. . . 4 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) ∈ dom card)
19		onsucb 7811	. . . . . 6 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
20		alephgeom 10132	. . . . . 6 ⊢ (suc 𝐴 ∈ On ↔ ω ⊆ (ℵ‘suc 𝐴))
21	19, 20	bitri 278	. . . . 5 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘suc 𝐴))
22		fvex 6886	. . . . . 6 ⊢ (ℵ‘suc 𝐴) ∈ V
23		ssdomg 9005	. . . . . 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 220	. . . 4 ⊢ (𝐴 ∈ On → ω ≼ (ℵ‘suc 𝐴))
26		alephordilem1 10123	. . . 4 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
27		infdif 10257	. . . 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 1398	. . 3 ⊢ (𝐴 ∈ On → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) ≈ (ℵ‘suc 𝐴))
29	15, 28	eqbrtrd 5126	. 2 ⊢ (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} ≈ (ℵ‘suc 𝐴))
30	29	ensymd 9010	1 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) ≈ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3412  Vcvv 3450   ∖ cdif 3895   ⊆ wss 3898   class class class wbr 5102  dom cdm 5647  Oncon0 6351  suc csuc 6353  ‘cfv 6527  ωcom 7860   ≈ cen 8948   ≼ cdom 8949   ≺ csdm 8950  cardccrd 9987  ℵcale 9988

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-inf2 9620

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-isom 6536  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-2o 8455  df-oadd 8458  df-er 8695  df-en 8952  df-dom 8953  df-sdom 8954  df-fin 8955  df-oi 9482  df-har 9529  df-dju 9953  df-card 9991  df-aleph 9992

This theorem is used by: (None)