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Theorem alephsuc3 10623
Description: An alternate representation of a successor aleph. Compare alephsuc 10111 and alephsuc2 10123. Equality can be obtained by taking the card of the right-hand side then using alephcard 10113 and carden 10594. (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 10123 . . . . 5 (𝐴 ∈ On → (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)})
2 alephcard 10113 . . . . . . 7 (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
3 alephon 10112 . . . . . . . . 9 (ℵ‘𝐴) ∈ On
4 onenon 9992 . . . . . . . . 9 ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
53, 4ax-mp 5 . . . . . . . 8 (ℵ‘𝐴) ∈ dom card
6 cardval2 10034 . . . . . . . 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 2756 . . . . . 6 (ℵ‘𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}
98a1i 11 . . . . 5 (𝐴 ∈ On → (ℵ‘𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)})
101, 9difeq12d 4122 . . . 4 (𝐴 ∈ On → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) = ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}))
11 difrab 4310 . . . . 5 ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}) = {𝑥 ∈ On ∣ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))}
12 bren2 9014 . . . . . 6 (𝑥 ≈ (ℵ‘𝐴) ↔ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴)))
1312rabbii 3425 . . . . 5 {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} = {𝑥 ∈ On ∣ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))}
1411, 13eqtr4i 2757 . . . 4 ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}) = {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)}
1510, 14eqtr2di 2783 . . 3 (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} = ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)))
16 alephon 10112 . . . . 5 (ℵ‘suc 𝐴) ∈ On
17 onenon 9992 . . . . 5 ((ℵ‘suc 𝐴) ∈ On → (ℵ‘suc 𝐴) ∈ dom card)
1816, 17mp1i 13 . . . 4 (𝐴 ∈ On → (ℵ‘suc 𝐴) ∈ dom card)
19 onsucb 7826 . . . . . 6 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
20 alephgeom 10125 . . . . . 6 (suc 𝐴 ∈ On ↔ ω ⊆ (ℵ‘suc 𝐴))
2119, 20bitri 274 . . . . 5 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘suc 𝐴))
22 fvex 6914 . . . . . 6 (ℵ‘suc 𝐴) ∈ V
23 ssdomg 9031 . . . . . 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 216 . . . 4 (𝐴 ∈ On → ω ≼ (ℵ‘suc 𝐴))
26 alephordilem1 10116 . . . 4 (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
27 infdif 10252 . . . 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 1368 . . 3 (𝐴 ∈ On → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) ≈ (ℵ‘suc 𝐴))
2915, 28eqbrtrd 5175 . 2 (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} ≈ (ℵ‘suc 𝐴))
3029ensymd 9036 1 (𝐴 ∈ On → (ℵ‘suc 𝐴) ≈ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1534  wcel 2099  {crab 3419  Vcvv 3462  cdif 3944  wss 3947   class class class wbr 5153  dom cdm 5682  Oncon0 6376  suc csuc 6378  cfv 6554  ωcom 7876  cen 8971  cdom 8972  csdm 8973  cardccrd 9978  cale 9979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9684
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-2o 8497  df-oadd 8500  df-er 8734  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-oi 9553  df-har 9600  df-dju 9944  df-card 9982  df-aleph 9983
This theorem is referenced by: (None)
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