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Theorem alephsuc3 9987
 Description: An alternate representation of a successor aleph. Compare alephsuc 9479 and alephsuc2 9491. Equality can be obtained by taking the card of the right-hand side then using alephcard 9481 and carden 9958. (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 9491 . . . . 5 (𝐴 ∈ On → (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)})
2 alephcard 9481 . . . . . . 7 (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
3 alephon 9480 . . . . . . . . 9 (ℵ‘𝐴) ∈ On
4 onenon 9362 . . . . . . . . 9 ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
53, 4ax-mp 5 . . . . . . . 8 (ℵ‘𝐴) ∈ dom card
6 cardval2 9404 . . . . . . . 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 2849 . . . . . 6 (ℵ‘𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}
98a1i 11 . . . . 5 (𝐴 ∈ On → (ℵ‘𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)})
101, 9difeq12d 4084 . . . 4 (𝐴 ∈ On → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) = ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}))
11 difrab 4260 . . . . 5 ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}) = {𝑥 ∈ On ∣ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))}
12 bren2 8523 . . . . . 6 (𝑥 ≈ (ℵ‘𝐴) ↔ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴)))
1312rabbii 3458 . . . . 5 {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} = {𝑥 ∈ On ∣ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))}
1411, 13eqtr4i 2850 . . . 4 ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}) = {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)}
1510, 14syl6req 2876 . . 3 (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} = ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)))
16 alephon 9480 . . . . 5 (ℵ‘suc 𝐴) ∈ On
17 onenon 9362 . . . . 5 ((ℵ‘suc 𝐴) ∈ On → (ℵ‘suc 𝐴) ∈ dom card)
1816, 17mp1i 13 . . . 4 (𝐴 ∈ On → (ℵ‘suc 𝐴) ∈ dom card)
19 sucelon 7515 . . . . . 6 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
20 alephgeom 9493 . . . . . 6 (suc 𝐴 ∈ On ↔ ω ⊆ (ℵ‘suc 𝐴))
2119, 20bitri 278 . . . . 5 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘suc 𝐴))
22 fvex 6664 . . . . . 6 (ℵ‘suc 𝐴) ∈ V
23 ssdomg 8538 . . . . . 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 220 . . . 4 (𝐴 ∈ On → ω ≼ (ℵ‘suc 𝐴))
26 alephordilem1 9484 . . . 4 (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
27 infdif 9616 . . . 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 5069 . 2 (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} ≈ (ℵ‘suc 𝐴))
3029ensymd 8543 1 (𝐴 ∈ On → (ℵ‘suc 𝐴) ≈ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2115  {crab 3136  Vcvv 3479   ∖ cdif 3915   ⊆ wss 3918   class class class wbr 5047  dom cdm 5536  Oncon0 6172  suc csuc 6174  ‘cfv 6336  ωcom 7563   ≈ cen 8489   ≼ cdom 8490   ≺ csdm 8491  cardccrd 9348  ℵcale 9349 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444  ax-inf2 9088 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rmo 3140  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-int 4858  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-se 5496  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-isom 6345  df-riota 7096  df-ov 7141  df-oprab 7142  df-mpo 7143  df-om 7564  df-1st 7672  df-2nd 7673  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-oi 8958  df-har 9005  df-dju 9314  df-card 9352  df-aleph 9353 This theorem is referenced by: (None)
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