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

Theorem alephsuc3 9349
Description: An alternate representation of a successor aleph. Compare alephsuc 8838 and alephsuc2 8850. Equality can be obtained by taking the card of the right-hand side then using alephcard 8840 and carden 9320. (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 8850 . . . . 5 (𝐴 ∈ On → (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)})
2 alephcard 8840 . . . . . . 7 (card‘(ℵ‘𝐴)) = (ℵ‘𝐴)
3 alephon 8839 . . . . . . . . 9 (ℵ‘𝐴) ∈ On
4 onenon 8722 . . . . . . . . 9 ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
53, 4ax-mp 5 . . . . . . . 8 (ℵ‘𝐴) ∈ dom card
6 cardval2 8764 . . . . . . . 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 2645 . . . . . 6 (ℵ‘𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}
98a1i 11 . . . . 5 (𝐴 ∈ On → (ℵ‘𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)})
101, 9difeq12d 3709 . . . 4 (𝐴 ∈ On → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) = ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}))
11 difrab 3879 . . . . 5 ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}) = {𝑥 ∈ On ∣ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))}
12 bren2 7933 . . . . . . 7 (𝑥 ≈ (ℵ‘𝐴) ↔ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴)))
1312a1i 11 . . . . . 6 (𝑥 ∈ On → (𝑥 ≈ (ℵ‘𝐴) ↔ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))))
1413rabbiia 3173 . . . . 5 {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} = {𝑥 ∈ On ∣ (𝑥 ≼ (ℵ‘𝐴) ∧ ¬ 𝑥 ≺ (ℵ‘𝐴))}
1511, 14eqtr4i 2646 . . . 4 ({𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)} ∖ {𝑥 ∈ On ∣ 𝑥 ≺ (ℵ‘𝐴)}) = {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)}
1610, 15syl6req 2672 . . 3 (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} = ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)))
17 alephon 8839 . . . . 5 (ℵ‘suc 𝐴) ∈ On
18 onenon 8722 . . . . 5 ((ℵ‘suc 𝐴) ∈ On → (ℵ‘suc 𝐴) ∈ dom card)
1917, 18mp1i 13 . . . 4 (𝐴 ∈ On → (ℵ‘suc 𝐴) ∈ dom card)
20 sucelon 6967 . . . . . 6 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
21 alephgeom 8852 . . . . . 6 (suc 𝐴 ∈ On ↔ ω ⊆ (ℵ‘suc 𝐴))
2220, 21bitri 264 . . . . 5 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘suc 𝐴))
23 fvex 6160 . . . . . 6 (ℵ‘suc 𝐴) ∈ V
24 ssdomg 7948 . . . . . 6 ((ℵ‘suc 𝐴) ∈ V → (ω ⊆ (ℵ‘suc 𝐴) → ω ≼ (ℵ‘suc 𝐴)))
2523, 24ax-mp 5 . . . . 5 (ω ⊆ (ℵ‘suc 𝐴) → ω ≼ (ℵ‘suc 𝐴))
2622, 25sylbi 207 . . . 4 (𝐴 ∈ On → ω ≼ (ℵ‘suc 𝐴))
27 alephordilem1 8843 . . . 4 (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
28 infdif 8978 . . . 4 (((ℵ‘suc 𝐴) ∈ dom card ∧ ω ≼ (ℵ‘suc 𝐴) ∧ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)) → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) ≈ (ℵ‘suc 𝐴))
2919, 26, 27, 28syl3anc 1323 . . 3 (𝐴 ∈ On → ((ℵ‘suc 𝐴) ∖ (ℵ‘𝐴)) ≈ (ℵ‘suc 𝐴))
3016, 29eqbrtrd 4637 . 2 (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)} ≈ (ℵ‘suc 𝐴))
3130ensymd 7954 1 (𝐴 ∈ On → (ℵ‘suc 𝐴) ≈ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3186  cdif 3553  wss 3556   class class class wbr 4615  dom cdm 5076  Oncon0 5684  suc csuc 5686  cfv 5849  ωcom 7015  cen 7899  cdom 7900  csdm 7901  cardccrd 8708  cale 8709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-inf2 8485
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-se 5036  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-isom 5858  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-2o 7509  df-oadd 7512  df-er 7690  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-oi 8362  df-har 8410  df-card 8712  df-aleph 8713  df-cda 8937
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator