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Theorem alephfp 8891
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 8892 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 8890 . 2 (𝐻 “ ω) ∈ ran ℵ
3 isinfcard 8875 . . 3 ((ω ⊆ (𝐻 “ ω) ∧ (card‘ (𝐻 “ ω)) = (𝐻 “ ω)) ↔ (𝐻 “ ω) ∈ ran ℵ)
4 cardalephex 8873 . . . 4 (ω ⊆ (𝐻 “ ω) → ((card‘ (𝐻 “ ω)) = (𝐻 “ ω) ↔ ∃𝑧 ∈ On (𝐻 “ ω) = (ℵ‘𝑧)))
54biimpa 501 . . 3 ((ω ⊆ (𝐻 “ ω) ∧ (card‘ (𝐻 “ ω)) = (𝐻 “ ω)) → ∃𝑧 ∈ On (𝐻 “ ω) = (ℵ‘𝑧))
63, 5sylbir 225 . 2 ( (𝐻 “ ω) ∈ ran ℵ → ∃𝑧 ∈ On (𝐻 “ ω) = (ℵ‘𝑧))
7 alephle 8871 . . . . . . . . 9 (𝑧 ∈ On → 𝑧 ⊆ (ℵ‘𝑧))
8 alephon 8852 . . . . . . . . . . 11 (ℵ‘𝑧) ∈ On
98onirri 5803 . . . . . . . . . 10 ¬ (ℵ‘𝑧) ∈ (ℵ‘𝑧)
10 frfnom 7490 . . . . . . . . . . . . . 14 (rec(ℵ, ω) ↾ ω) Fn ω
111fneq1i 5953 . . . . . . . . . . . . . 14 (𝐻 Fn ω ↔ (rec(ℵ, ω) ↾ ω) Fn ω)
1210, 11mpbir 221 . . . . . . . . . . . . 13 𝐻 Fn ω
13 fnfun 5956 . . . . . . . . . . . . 13 (𝐻 Fn ω → Fun 𝐻)
14 eluniima 6473 . . . . . . . . . . . . 13 (Fun 𝐻 → (𝑧 (𝐻 “ ω) ↔ ∃𝑣 ∈ ω 𝑧 ∈ (𝐻𝑣)))
1512, 13, 14mp2b 10 . . . . . . . . . . . 12 (𝑧 (𝐻 “ ω) ↔ ∃𝑣 ∈ ω 𝑧 ∈ (𝐻𝑣))
16 alephsson 8883 . . . . . . . . . . . . . . . 16 ran ℵ ⊆ On
171alephfplem3 8889 . . . . . . . . . . . . . . . 16 (𝑣 ∈ ω → (𝐻𝑣) ∈ ran ℵ)
1816, 17sseldi 3586 . . . . . . . . . . . . . . 15 (𝑣 ∈ ω → (𝐻𝑣) ∈ On)
19 alephord2i 8860 . . . . . . . . . . . . . . 15 ((𝐻𝑣) ∈ On → (𝑧 ∈ (𝐻𝑣) → (ℵ‘𝑧) ∈ (ℵ‘(𝐻𝑣))))
2018, 19syl 17 . . . . . . . . . . . . . 14 (𝑣 ∈ ω → (𝑧 ∈ (𝐻𝑣) → (ℵ‘𝑧) ∈ (ℵ‘(𝐻𝑣))))
211alephfplem2 8888 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ ω → (𝐻‘suc 𝑣) = (ℵ‘(𝐻𝑣)))
22 peano2 7048 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ ω → suc 𝑣 ∈ ω)
23 fnfvelrn 6322 . . . . . . . . . . . . . . . . . . . 20 ((𝐻 Fn ω ∧ suc 𝑣 ∈ ω) → (𝐻‘suc 𝑣) ∈ ran 𝐻)
2412, 23mpan 705 . . . . . . . . . . . . . . . . . . 19 (suc 𝑣 ∈ ω → (𝐻‘suc 𝑣) ∈ ran 𝐻)
25 fnima 5977 . . . . . . . . . . . . . . . . . . . 20 (𝐻 Fn ω → (𝐻 “ ω) = ran 𝐻)
2612, 25ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝐻 “ ω) = ran 𝐻
2724, 26syl6eleqr 2709 . . . . . . . . . . . . . . . . . 18 (suc 𝑣 ∈ ω → (𝐻‘suc 𝑣) ∈ (𝐻 “ ω))
2822, 27syl 17 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ ω → (𝐻‘suc 𝑣) ∈ (𝐻 “ ω))
2921, 28eqeltrrd 2699 . . . . . . . . . . . . . . . 16 (𝑣 ∈ ω → (ℵ‘(𝐻𝑣)) ∈ (𝐻 “ ω))
30 elssuni 4440 . . . . . . . . . . . . . . . 16 ((ℵ‘(𝐻𝑣)) ∈ (𝐻 “ ω) → (ℵ‘(𝐻𝑣)) ⊆ (𝐻 “ ω))
3129, 30syl 17 . . . . . . . . . . . . . . 15 (𝑣 ∈ ω → (ℵ‘(𝐻𝑣)) ⊆ (𝐻 “ ω))
3231sseld 3587 . . . . . . . . . . . . . 14 (𝑣 ∈ ω → ((ℵ‘𝑧) ∈ (ℵ‘(𝐻𝑣)) → (ℵ‘𝑧) ∈ (𝐻 “ ω)))
3320, 32syld 47 . . . . . . . . . . . . 13 (𝑣 ∈ ω → (𝑧 ∈ (𝐻𝑣) → (ℵ‘𝑧) ∈ (𝐻 “ ω)))
3433rexlimiv 3022 . . . . . . . . . . . 12 (∃𝑣 ∈ ω 𝑧 ∈ (𝐻𝑣) → (ℵ‘𝑧) ∈ (𝐻 “ ω))
3515, 34sylbi 207 . . . . . . . . . . 11 (𝑧 (𝐻 “ ω) → (ℵ‘𝑧) ∈ (𝐻 “ ω))
36 eleq2 2687 . . . . . . . . . . . 12 ( (𝐻 “ ω) = (ℵ‘𝑧) → (𝑧 (𝐻 “ ω) ↔ 𝑧 ∈ (ℵ‘𝑧)))
37 eleq2 2687 . . . . . . . . . . . 12 ( (𝐻 “ ω) = (ℵ‘𝑧) → ((ℵ‘𝑧) ∈ (𝐻 “ ω) ↔ (ℵ‘𝑧) ∈ (ℵ‘𝑧)))
3836, 37imbi12d 334 . . . . . . . . . . 11 ( (𝐻 “ ω) = (ℵ‘𝑧) → ((𝑧 (𝐻 “ ω) → (ℵ‘𝑧) ∈ (𝐻 “ ω)) ↔ (𝑧 ∈ (ℵ‘𝑧) → (ℵ‘𝑧) ∈ (ℵ‘𝑧))))
3935, 38mpbii 223 . . . . . . . . . 10 ( (𝐻 “ ω) = (ℵ‘𝑧) → (𝑧 ∈ (ℵ‘𝑧) → (ℵ‘𝑧) ∈ (ℵ‘𝑧)))
409, 39mtoi 190 . . . . . . . . 9 ( (𝐻 “ ω) = (ℵ‘𝑧) → ¬ 𝑧 ∈ (ℵ‘𝑧))
417, 40anim12i 589 . . . . . . . 8 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧)))
42 eloni 5702 . . . . . . . . . 10 (𝑧 ∈ On → Ord 𝑧)
438onordi 5801 . . . . . . . . . 10 Ord (ℵ‘𝑧)
44 ordtri4 5730 . . . . . . . . . 10 ((Ord 𝑧 ∧ Ord (ℵ‘𝑧)) → (𝑧 = (ℵ‘𝑧) ↔ (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧))))
4542, 43, 44sylancl 693 . . . . . . . . 9 (𝑧 ∈ On → (𝑧 = (ℵ‘𝑧) ↔ (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧))))
4645adantr 481 . . . . . . . 8 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → (𝑧 = (ℵ‘𝑧) ↔ (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧))))
4741, 46mpbird 247 . . . . . . 7 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → 𝑧 = (ℵ‘𝑧))
48 eqeq2 2632 . . . . . . . 8 ( (𝐻 “ ω) = (ℵ‘𝑧) → (𝑧 = (𝐻 “ ω) ↔ 𝑧 = (ℵ‘𝑧)))
4948adantl 482 . . . . . . 7 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → (𝑧 = (𝐻 “ ω) ↔ 𝑧 = (ℵ‘𝑧)))
5047, 49mpbird 247 . . . . . 6 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → 𝑧 = (𝐻 “ ω))
5150eqcomd 2627 . . . . 5 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → (𝐻 “ ω) = 𝑧)
5251fveq2d 6162 . . . 4 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → (ℵ‘ (𝐻 “ ω)) = (ℵ‘𝑧))
53 eqeq2 2632 . . . . 5 ( (𝐻 “ ω) = (ℵ‘𝑧) → ((ℵ‘ (𝐻 “ ω)) = (𝐻 “ ω) ↔ (ℵ‘ (𝐻 “ ω)) = (ℵ‘𝑧)))
5453adantl 482 . . . 4 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → ((ℵ‘ (𝐻 “ ω)) = (𝐻 “ ω) ↔ (ℵ‘ (𝐻 “ ω)) = (ℵ‘𝑧)))
5552, 54mpbird 247 . . 3 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → (ℵ‘ (𝐻 “ ω)) = (𝐻 “ ω))
5655rexlimiva 3023 . 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 196  wa 384   = wceq 1480  wcel 1987  wrex 2909  wss 3560   cuni 4409  ran crn 5085  cres 5086  cima 5087  Ord word 5691  Oncon0 5692  suc csuc 5694  Fun wfun 5851   Fn wfn 5852  cfv 5857  ωcom 7027  reccrdg 7465  cardccrd 8721  cale 8722
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 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498
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 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-om 7028  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-oi 8375  df-har 8423  df-card 8725  df-aleph 8726
This theorem is referenced by:  alephfp2  8892
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