MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  alephsuc3 Structured version   Visualization version   GIF version

Theorem alephsuc3 9615
Description: An alternate representation of a successor aleph. Compare alephsuc 9102 and alephsuc2 9114. Equality can be obtained by taking the card of the right-hand side then using alephcard 9104 and carden 9586. (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
StepHypRef Expression
1 alephsuc2 9114 . . . . 5 (𝐴 ∈ On → (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)})
2 alephcard 9104 . . . . . . 7 (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
3 alephon 9103 . . . . . . . . 9 (ℵ‘𝐴) ∈ On
4 onenon 8986 . . . . . . . . 9 ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
53, 4ax-mp 5 . . . . . . . 8 (ℵ‘𝐴) ∈ dom card
6 cardval2 9028 . . . . . . . 8 ((ℵ‘𝐴) ∈ dom card → (card‘(ℵ‘𝐴)) = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)})
75, 6ax-mp 5 . . . . . . 7 (card‘(ℵ‘𝐴)) = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}
82, 7eqtr3i 2785 . . . . . 6 (ℵ‘𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}
98a1i 11 . . . . 5 (𝐴 ∈ On → (ℵ‘𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)})
101, 9difeq12d 3873 . . . 4 (𝐴 ∈ On → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) = ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}))
11 difrab 4045 . . . . 5 ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}) = {𝑥 ∈ On ∣ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))}
12 bren2 8155 . . . . . 6 (𝑥 ≈ (ℵ‘𝐴) ↔ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴)))
1312rabbii 3326 . . . . 5 {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} = {𝑥 ∈ On ∣ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))}
1411, 13eqtr4i 2786 . . . 4 ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}) = {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)}
1510, 14syl6req 2812 . . 3 (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} = ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)))
16 alephon 9103 . . . . 5 (ℵ‘suc 𝐴) ∈ On
17 onenon 8986 . . . . 5 ((ℵ‘suc 𝐴) ∈ On → (ℵ‘suc 𝐴) ∈ dom card)
1816, 17mp1i 13 . . . 4 (𝐴 ∈ On → (ℵ‘suc 𝐴) ∈ dom card)
19 sucelon 7184 . . . . . 6 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
20 alephgeom 9116 . . . . . 6 (suc 𝐴 ∈ On ↔ ω ⊆ (ℵ‘suc 𝐴))
2119, 20bitri 264 . . . . 5 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘suc 𝐴))
22 fvex 6364 . . . . . 6 (ℵ‘suc 𝐴) ∈ V
23 ssdomg 8170 . . . . . 6 ((ℵ‘suc 𝐴) ∈ V → (ω ⊆ (ℵ‘suc 𝐴) → ω ≼ (ℵ‘suc 𝐴)))
2422, 23ax-mp 5 . . . . 5 (ω ⊆ (ℵ‘suc 𝐴) → ω ≼ (ℵ‘suc 𝐴))
2521, 24sylbi 207 . . . 4 (𝐴 ∈ On → ω ≼ (ℵ‘suc 𝐴))
26 alephordilem1 9107 . . . 4 (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
27 infdif 9244 . . . 4 (((ℵ‘suc 𝐴) ∈ dom card ∧ ω ≼ (ℵ‘suc 𝐴) ∧ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)) → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) ≈ (ℵ‘suc 𝐴))
2818, 25, 26, 27syl3anc 1477 . . 3 (𝐴 ∈ On → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) ≈ (ℵ‘suc 𝐴))
2915, 28eqbrtrd 4827 . 2 (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} ≈ (ℵ‘suc 𝐴))
3029ensymd 8175 1 (𝐴 ∈ On → (ℵ‘suc 𝐴) ≈ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1632  wcel 2140  {crab 3055  Vcvv 3341  cdif 3713  wss 3716   class class class wbr 4805  dom cdm 5267  Oncon0 5885  suc csuc 5887  cfv 6050  ωcom 7232  cen 8121  cdom 8122  csdm 8123  cardccrd 8972  cale 8973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-inf2 8714
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-int 4629  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-se 5227  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-isom 6059  df-riota 6776  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-om 7233  df-1st 7335  df-2nd 7336  df-wrecs 7578  df-recs 7639  df-rdg 7677  df-1o 7731  df-2o 7732  df-oadd 7735  df-er 7914  df-en 8125  df-dom 8126  df-sdom 8127  df-fin 8128  df-oi 8583  df-har 8631  df-card 8976  df-aleph 8977  df-cda 9203
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator