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Theorem alephsuc3 9724
Description: An alternate representation of a successor aleph. Compare alephsuc 9211 and alephsuc2 9223. Equality can be obtained by taking the card of the right-hand side then using alephcard 9213 and carden 9695. (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 9223 . . . . 5 (𝐴 ∈ On → (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)})
2 alephcard 9213 . . . . . . 7 (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
3 alephon 9212 . . . . . . . . 9 (ℵ‘𝐴) ∈ On
4 onenon 9095 . . . . . . . . 9 ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
53, 4ax-mp 5 . . . . . . . 8 (ℵ‘𝐴) ∈ dom card
6 cardval2 9137 . . . . . . . 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 2851 . . . . . 6 (ℵ‘𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}
98a1i 11 . . . . 5 (𝐴 ∈ On → (ℵ‘𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)})
101, 9difeq12d 3958 . . . 4 (𝐴 ∈ On → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) = ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}))
11 difrab 4132 . . . . 5 ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}) = {𝑥 ∈ On ∣ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))}
12 bren2 8259 . . . . . 6 (𝑥 ≈ (ℵ‘𝐴) ↔ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴)))
1312rabbii 3398 . . . . 5 {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} = {𝑥 ∈ On ∣ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))}
1411, 13eqtr4i 2852 . . . 4 ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}) = {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)}
1510, 14syl6req 2878 . . 3 (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} = ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)))
16 alephon 9212 . . . . 5 (ℵ‘suc 𝐴) ∈ On
17 onenon 9095 . . . . 5 ((ℵ‘suc 𝐴) ∈ On → (ℵ‘suc 𝐴) ∈ dom card)
1816, 17mp1i 13 . . . 4 (𝐴 ∈ On → (ℵ‘suc 𝐴) ∈ dom card)
19 sucelon 7283 . . . . . 6 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
20 alephgeom 9225 . . . . . 6 (suc 𝐴 ∈ On ↔ ω ⊆ (ℵ‘suc 𝐴))
2119, 20bitri 267 . . . . 5 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘suc 𝐴))
22 fvex 6450 . . . . . 6 (ℵ‘suc 𝐴) ∈ V
23 ssdomg 8274 . . . . . 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 209 . . . 4 (𝐴 ∈ On → ω ≼ (ℵ‘suc 𝐴))
26 alephordilem1 9216 . . . 4 (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
27 infdif 9353 . . . 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 1494 . . 3 (𝐴 ∈ On → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) ≈ (ℵ‘suc 𝐴))
2915, 28eqbrtrd 4897 . 2 (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} ≈ (ℵ‘suc 𝐴))
3029ensymd 8279 1 (𝐴 ∈ On → (ℵ‘suc 𝐴) ≈ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386   = wceq 1656  wcel 2164  {crab 3121  Vcvv 3414  cdif 3795  wss 3798   class class class wbr 4875  dom cdm 5346  Oncon0 5967  suc csuc 5969  cfv 6127  ωcom 7331  cen 8225  cdom 8226  csdm 8227  cardccrd 9081  cale 9082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-inf2 8822
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-se 5306  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-isom 6136  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-2o 7832  df-oadd 7835  df-er 8014  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-oi 8691  df-har 8739  df-card 9085  df-aleph 9086  df-cda 9312
This theorem is referenced by: (None)
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