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Theorem alephfp 9131
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 9132 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 9130 . 2 (𝐻 “ ω) ∈ ran ℵ
3 isinfcard 9115 . . 3 ((ω ⊆ (𝐻 “ ω) ∧ (card‘ (𝐻 “ ω)) = (𝐻 “ ω)) ↔ (𝐻 “ ω) ∈ ran ℵ)
4 cardalephex 9113 . . . 4 (ω ⊆ (𝐻 “ ω) → ((card‘ (𝐻 “ ω)) = (𝐻 “ ω) ↔ ∃𝑧 ∈ On (𝐻 “ ω) = (ℵ‘𝑧)))
54biimpa 462 . . 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 9111 . . . . . . . . 9 (𝑧 ∈ On → 𝑧 ⊆ (ℵ‘𝑧))
8 alephon 9092 . . . . . . . . . . 11 (ℵ‘𝑧) ∈ On
98onirri 5977 . . . . . . . . . 10 ¬ (ℵ‘𝑧) ∈ (ℵ‘𝑧)
10 frfnom 7683 . . . . . . . . . . . . . 14 (rec(ℵ, ω) ↾ ω) Fn ω
111fneq1i 6125 . . . . . . . . . . . . . 14 (𝐻 Fn ω ↔ (rec(ℵ, ω) ↾ ω) Fn ω)
1210, 11mpbir 221 . . . . . . . . . . . . 13 𝐻 Fn ω
13 fnfun 6128 . . . . . . . . . . . . 13 (𝐻 Fn ω → Fun 𝐻)
14 eluniima 6651 . . . . . . . . . . . . 13 (Fun 𝐻 → (𝑧 (𝐻 “ ω) ↔ ∃𝑣 ∈ ω 𝑧 ∈ (𝐻𝑣)))
1512, 13, 14mp2b 10 . . . . . . . . . . . 12 (𝑧 (𝐻 “ ω) ↔ ∃𝑣 ∈ ω 𝑧 ∈ (𝐻𝑣))
16 alephsson 9123 . . . . . . . . . . . . . . . 16 ran ℵ ⊆ On
171alephfplem3 9129 . . . . . . . . . . . . . . . 16 (𝑣 ∈ ω → (𝐻𝑣) ∈ ran ℵ)
1816, 17sseldi 3750 . . . . . . . . . . . . . . 15 (𝑣 ∈ ω → (𝐻𝑣) ∈ On)
19 alephord2i 9100 . . . . . . . . . . . . . . 15 ((𝐻𝑣) ∈ On → (𝑧 ∈ (𝐻𝑣) → (ℵ‘𝑧) ∈ (ℵ‘(𝐻𝑣))))
2018, 19syl 17 . . . . . . . . . . . . . 14 (𝑣 ∈ ω → (𝑧 ∈ (𝐻𝑣) → (ℵ‘𝑧) ∈ (ℵ‘(𝐻𝑣))))
211alephfplem2 9128 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ ω → (𝐻‘suc 𝑣) = (ℵ‘(𝐻𝑣)))
22 peano2 7233 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ ω → suc 𝑣 ∈ ω)
23 fnfvelrn 6499 . . . . . . . . . . . . . . . . . . . 20 ((𝐻 Fn ω ∧ suc 𝑣 ∈ ω) → (𝐻‘suc 𝑣) ∈ ran 𝐻)
2412, 23mpan 670 . . . . . . . . . . . . . . . . . . 19 (suc 𝑣 ∈ ω → (𝐻‘suc 𝑣) ∈ ran 𝐻)
25 fnima 6150 . . . . . . . . . . . . . . . . . . . 20 (𝐻 Fn ω → (𝐻 “ ω) = ran 𝐻)
2612, 25ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝐻 “ ω) = ran 𝐻
2724, 26syl6eleqr 2861 . . . . . . . . . . . . . . . . . 18 (suc 𝑣 ∈ ω → (𝐻‘suc 𝑣) ∈ (𝐻 “ ω))
2822, 27syl 17 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ ω → (𝐻‘suc 𝑣) ∈ (𝐻 “ ω))
2921, 28eqeltrrd 2851 . . . . . . . . . . . . . . . 16 (𝑣 ∈ ω → (ℵ‘(𝐻𝑣)) ∈ (𝐻 “ ω))
30 elssuni 4603 . . . . . . . . . . . . . . . 16 ((ℵ‘(𝐻𝑣)) ∈ (𝐻 “ ω) → (ℵ‘(𝐻𝑣)) ⊆ (𝐻 “ ω))
3129, 30syl 17 . . . . . . . . . . . . . . 15 (𝑣 ∈ ω → (ℵ‘(𝐻𝑣)) ⊆ (𝐻 “ ω))
3231sseld 3751 . . . . . . . . . . . . . 14 (𝑣 ∈ ω → ((ℵ‘𝑧) ∈ (ℵ‘(𝐻𝑣)) → (ℵ‘𝑧) ∈ (𝐻 “ ω)))
3320, 32syld 47 . . . . . . . . . . . . 13 (𝑣 ∈ ω → (𝑧 ∈ (𝐻𝑣) → (ℵ‘𝑧) ∈ (𝐻 “ ω)))
3433rexlimiv 3175 . . . . . . . . . . . 12 (∃𝑣 ∈ ω 𝑧 ∈ (𝐻𝑣) → (ℵ‘𝑧) ∈ (𝐻 “ ω))
3515, 34sylbi 207 . . . . . . . . . . 11 (𝑧 (𝐻 “ ω) → (ℵ‘𝑧) ∈ (𝐻 “ ω))
36 eleq2 2839 . . . . . . . . . . . 12 ( (𝐻 “ ω) = (ℵ‘𝑧) → (𝑧 (𝐻 “ ω) ↔ 𝑧 ∈ (ℵ‘𝑧)))
37 eleq2 2839 . . . . . . . . . . . 12 ( (𝐻 “ ω) = (ℵ‘𝑧) → ((ℵ‘𝑧) ∈ (𝐻 “ ω) ↔ (ℵ‘𝑧) ∈ (ℵ‘𝑧)))
3836, 37imbi12d 333 . . . . . . . . . . 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 600 . . . . . . . 8 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧)))
42 eloni 5876 . . . . . . . . . 10 (𝑧 ∈ On → Ord 𝑧)
438onordi 5975 . . . . . . . . . 10 Ord (ℵ‘𝑧)
44 ordtri4 5904 . . . . . . . . . 10 ((Ord 𝑧 ∧ Ord (ℵ‘𝑧)) → (𝑧 = (ℵ‘𝑧) ↔ (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧))))
4542, 43, 44sylancl 574 . . . . . . . . 9 (𝑧 ∈ On → (𝑧 = (ℵ‘𝑧) ↔ (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧))))
4645adantr 466 . . . . . . . 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 2782 . . . . . . . 8 ( (𝐻 “ ω) = (ℵ‘𝑧) → (𝑧 = (𝐻 “ ω) ↔ 𝑧 = (ℵ‘𝑧)))
4948adantl 467 . . . . . . 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 2777 . . . . 5 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → (𝐻 “ ω) = 𝑧)
5251fveq2d 6336 . . . 4 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → (ℵ‘ (𝐻 “ ω)) = (ℵ‘𝑧))
53 eqeq2 2782 . . . . 5 ( (𝐻 “ ω) = (ℵ‘𝑧) → ((ℵ‘ (𝐻 “ ω)) = (𝐻 “ ω) ↔ (ℵ‘ (𝐻 “ ω)) = (ℵ‘𝑧)))
5453adantl 467 . . . 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 3176 . 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 382   = wceq 1631  wcel 2145  wrex 3062  wss 3723   cuni 4574  ran crn 5250  cres 5251  cima 5252  Ord word 5865  Oncon0 5866  suc csuc 5868  Fun wfun 6025   Fn wfn 6026  cfv 6031  ωcom 7212  reccrdg 7658  cardccrd 8961  cale 8962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-oi 8571  df-har 8619  df-card 8965  df-aleph 8966
This theorem is referenced by:  alephfp2  9132
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