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Theorem alephfp 8788
Description: The aleph function has a fixed point. Similar to Proposition 11.18 of [TakeutiZaring] p. 104, except that we construct an actual example of a fixed point rather than just showing its existence. See alephfp2 8789 for an abbreviated version just showing existence. (Contributed by NM, 6-Nov-2004.) (Proof shortened by Mario Carneiro, 15-May-2015.)
Hypothesis
Ref Expression
alephfplem.1 𝐻 = (rec(ℵ, ω) ↾ ω)
Assertion
Ref Expression
alephfp (ℵ‘ (𝐻 “ ω)) = (𝐻 “ ω)

Proof of Theorem alephfp
Dummy variables 𝑧 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 alephfplem.1 . . 3 𝐻 = (rec(ℵ, ω) ↾ ω)
21alephfplem4 8787 . 2 (𝐻 “ ω) ∈ ran ℵ
3 isinfcard 8772 . . 3 ((ω ⊆ (𝐻 “ ω) ∧ (card‘ (𝐻 “ ω)) = (𝐻 “ ω)) ↔ (𝐻 “ ω) ∈ ran ℵ)
4 cardalephex 8770 . . . 4 (ω ⊆ (𝐻 “ ω) → ((card‘ (𝐻 “ ω)) = (𝐻 “ ω) ↔ ∃𝑧 ∈ On (𝐻 “ ω) = (ℵ‘𝑧)))
54biimpa 499 . . 3 ((ω ⊆ (𝐻 “ ω) ∧ (card‘ (𝐻 “ ω)) = (𝐻 “ ω)) → ∃𝑧 ∈ On (𝐻 “ ω) = (ℵ‘𝑧))
63, 5sylbir 223 . 2 ( (𝐻 “ ω) ∈ ran ℵ → ∃𝑧 ∈ On (𝐻 “ ω) = (ℵ‘𝑧))
7 alephle 8768 . . . . . . . . 9 (𝑧 ∈ On → 𝑧 ⊆ (ℵ‘𝑧))
8 alephon 8749 . . . . . . . . . . 11 (ℵ‘𝑧) ∈ On
98onirri 5734 . . . . . . . . . 10 ¬ (ℵ‘𝑧) ∈ (ℵ‘𝑧)
10 frfnom 7391 . . . . . . . . . . . . . 14 (rec(ℵ, ω) ↾ ω) Fn ω
111fneq1i 5882 . . . . . . . . . . . . . 14 (𝐻 Fn ω ↔ (rec(ℵ, ω) ↾ ω) Fn ω)
1210, 11mpbir 219 . . . . . . . . . . . . 13 𝐻 Fn ω
13 fnfun 5885 . . . . . . . . . . . . 13 (𝐻 Fn ω → Fun 𝐻)
14 eluniima 6387 . . . . . . . . . . . . 13 (Fun 𝐻 → (𝑧 (𝐻 “ ω) ↔ ∃𝑣 ∈ ω 𝑧 ∈ (𝐻𝑣)))
1512, 13, 14mp2b 10 . . . . . . . . . . . 12 (𝑧 (𝐻 “ ω) ↔ ∃𝑣 ∈ ω 𝑧 ∈ (𝐻𝑣))
16 alephsson 8780 . . . . . . . . . . . . . . . 16 ran ℵ ⊆ On
171alephfplem3 8786 . . . . . . . . . . . . . . . 16 (𝑣 ∈ ω → (𝐻𝑣) ∈ ran ℵ)
1816, 17sseldi 3562 . . . . . . . . . . . . . . 15 (𝑣 ∈ ω → (𝐻𝑣) ∈ On)
19 alephord2i 8757 . . . . . . . . . . . . . . 15 ((𝐻𝑣) ∈ On → (𝑧 ∈ (𝐻𝑣) → (ℵ‘𝑧) ∈ (ℵ‘(𝐻𝑣))))
2018, 19syl 17 . . . . . . . . . . . . . 14 (𝑣 ∈ ω → (𝑧 ∈ (𝐻𝑣) → (ℵ‘𝑧) ∈ (ℵ‘(𝐻𝑣))))
211alephfplem2 8785 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ ω → (𝐻‘suc 𝑣) = (ℵ‘(𝐻𝑣)))
22 peano2 6952 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ ω → suc 𝑣 ∈ ω)
23 fnfvelrn 6246 . . . . . . . . . . . . . . . . . . . 20 ((𝐻 Fn ω ∧ suc 𝑣 ∈ ω) → (𝐻‘suc 𝑣) ∈ ran 𝐻)
2412, 23mpan 701 . . . . . . . . . . . . . . . . . . 19 (suc 𝑣 ∈ ω → (𝐻‘suc 𝑣) ∈ ran 𝐻)
25 fnima 5906 . . . . . . . . . . . . . . . . . . . 20 (𝐻 Fn ω → (𝐻 “ ω) = ran 𝐻)
2612, 25ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝐻 “ ω) = ran 𝐻
2724, 26syl6eleqr 2695 . . . . . . . . . . . . . . . . . 18 (suc 𝑣 ∈ ω → (𝐻‘suc 𝑣) ∈ (𝐻 “ ω))
2822, 27syl 17 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ ω → (𝐻‘suc 𝑣) ∈ (𝐻 “ ω))
2921, 28eqeltrrd 2685 . . . . . . . . . . . . . . . 16 (𝑣 ∈ ω → (ℵ‘(𝐻𝑣)) ∈ (𝐻 “ ω))
30 elssuni 4394 . . . . . . . . . . . . . . . 16 ((ℵ‘(𝐻𝑣)) ∈ (𝐻 “ ω) → (ℵ‘(𝐻𝑣)) ⊆ (𝐻 “ ω))
3129, 30syl 17 . . . . . . . . . . . . . . 15 (𝑣 ∈ ω → (ℵ‘(𝐻𝑣)) ⊆ (𝐻 “ ω))
3231sseld 3563 . . . . . . . . . . . . . 14 (𝑣 ∈ ω → ((ℵ‘𝑧) ∈ (ℵ‘(𝐻𝑣)) → (ℵ‘𝑧) ∈ (𝐻 “ ω)))
3320, 32syld 45 . . . . . . . . . . . . 13 (𝑣 ∈ ω → (𝑧 ∈ (𝐻𝑣) → (ℵ‘𝑧) ∈ (𝐻 “ ω)))
3433rexlimiv 3005 . . . . . . . . . . . 12 (∃𝑣 ∈ ω 𝑧 ∈ (𝐻𝑣) → (ℵ‘𝑧) ∈ (𝐻 “ ω))
3515, 34sylbi 205 . . . . . . . . . . 11 (𝑧 (𝐻 “ ω) → (ℵ‘𝑧) ∈ (𝐻 “ ω))
36 eleq2 2673 . . . . . . . . . . . 12 ( (𝐻 “ ω) = (ℵ‘𝑧) → (𝑧 (𝐻 “ ω) ↔ 𝑧 ∈ (ℵ‘𝑧)))
37 eleq2 2673 . . . . . . . . . . . 12 ( (𝐻 “ ω) = (ℵ‘𝑧) → ((ℵ‘𝑧) ∈ (𝐻 “ ω) ↔ (ℵ‘𝑧) ∈ (ℵ‘𝑧)))
3836, 37imbi12d 332 . . . . . . . . . . 11 ( (𝐻 “ ω) = (ℵ‘𝑧) → ((𝑧 (𝐻 “ ω) → (ℵ‘𝑧) ∈ (𝐻 “ ω)) ↔ (𝑧 ∈ (ℵ‘𝑧) → (ℵ‘𝑧) ∈ (ℵ‘𝑧))))
3935, 38mpbii 221 . . . . . . . . . 10 ( (𝐻 “ ω) = (ℵ‘𝑧) → (𝑧 ∈ (ℵ‘𝑧) → (ℵ‘𝑧) ∈ (ℵ‘𝑧)))
409, 39mtoi 188 . . . . . . . . 9 ( (𝐻 “ ω) = (ℵ‘𝑧) → ¬ 𝑧 ∈ (ℵ‘𝑧))
417, 40anim12i 587 . . . . . . . 8 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧)))
42 eloni 5633 . . . . . . . . . 10 (𝑧 ∈ On → Ord 𝑧)
438onordi 5732 . . . . . . . . . 10 Ord (ℵ‘𝑧)
44 ordtri4 5661 . . . . . . . . . 10 ((Ord 𝑧 ∧ Ord (ℵ‘𝑧)) → (𝑧 = (ℵ‘𝑧) ↔ (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧))))
4542, 43, 44sylancl 692 . . . . . . . . 9 (𝑧 ∈ On → (𝑧 = (ℵ‘𝑧) ↔ (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧))))
4645adantr 479 . . . . . . . 8 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → (𝑧 = (ℵ‘𝑧) ↔ (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧))))
4741, 46mpbird 245 . . . . . . 7 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → 𝑧 = (ℵ‘𝑧))
48 eqeq2 2617 . . . . . . . 8 ( (𝐻 “ ω) = (ℵ‘𝑧) → (𝑧 = (𝐻 “ ω) ↔ 𝑧 = (ℵ‘𝑧)))
4948adantl 480 . . . . . . 7 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → (𝑧 = (𝐻 “ ω) ↔ 𝑧 = (ℵ‘𝑧)))
5047, 49mpbird 245 . . . . . 6 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → 𝑧 = (𝐻 “ ω))
5150eqcomd 2612 . . . . 5 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → (𝐻 “ ω) = 𝑧)
5251fveq2d 6089 . . . 4 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → (ℵ‘ (𝐻 “ ω)) = (ℵ‘𝑧))
53 eqeq2 2617 . . . . 5 ( (𝐻 “ ω) = (ℵ‘𝑧) → ((ℵ‘ (𝐻 “ ω)) = (𝐻 “ ω) ↔ (ℵ‘ (𝐻 “ ω)) = (ℵ‘𝑧)))
5453adantl 480 . . . 4 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → ((ℵ‘ (𝐻 “ ω)) = (𝐻 “ ω) ↔ (ℵ‘ (𝐻 “ ω)) = (ℵ‘𝑧)))
5552, 54mpbird 245 . . 3 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → (ℵ‘ (𝐻 “ ω)) = (𝐻 “ ω))
5655rexlimiva 3006 . 2 (∃𝑧 ∈ On (𝐻 “ ω) = (ℵ‘𝑧) → (ℵ‘ (𝐻 “ ω)) = (𝐻 “ ω))
572, 6, 56mp2b 10 1 (ℵ‘ (𝐻 “ ω)) = (𝐻 “ ω)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wrex 2893  wss 3536   cuni 4363  ran crn 5026  cres 5027  cima 5028  Ord word 5622  Oncon0 5623  suc csuc 5625  Fun wfun 5781   Fn wfn 5782  cfv 5787  ωcom 6931  reccrdg 7366  cardccrd 8618  cale 8619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-inf2 8395
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-se 4985  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-isom 5796  df-riota 6486  df-om 6932  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-er 7603  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-oi 8272  df-har 8320  df-card 8622  df-aleph 8623
This theorem is referenced by:  alephfp2  8789
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