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Theorem alephfp 10044
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 10045 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 10043 . 2 (𝐻 “ ω) ∈ ran ℵ
3 isinfcard 10028 . . 3 ((ω ⊆ (𝐻 “ ω) ∧ (card‘ (𝐻 “ ω)) = (𝐻 “ ω)) ↔ (𝐻 “ ω) ∈ ran ℵ)
4 cardalephex 10026 . . . 4 (ω ⊆ (𝐻 “ ω) → ((card‘ (𝐻 “ ω)) = (𝐻 “ ω) ↔ ∃𝑧 ∈ On (𝐻 “ ω) = (ℵ‘𝑧)))
54biimpa 477 . . 3 ((ω ⊆ (𝐻 “ ω) ∧ (card‘ (𝐻 “ ω)) = (𝐻 “ ω)) → ∃𝑧 ∈ On (𝐻 “ ω) = (ℵ‘𝑧))
63, 5sylbir 234 . 2 ( (𝐻 “ ω) ∈ ran ℵ → ∃𝑧 ∈ On (𝐻 “ ω) = (ℵ‘𝑧))
7 alephle 10024 . . . . . . . . 9 (𝑧 ∈ On → 𝑧 ⊆ (ℵ‘𝑧))
8 alephon 10005 . . . . . . . . . . 11 (ℵ‘𝑧) ∈ On
98onirri 6430 . . . . . . . . . 10 ¬ (ℵ‘𝑧) ∈ (ℵ‘𝑧)
10 frfnom 8381 . . . . . . . . . . . . . 14 (rec(ℵ, ω) ↾ ω) Fn ω
111fneq1i 6599 . . . . . . . . . . . . . 14 (𝐻 Fn ω ↔ (rec(ℵ, ω) ↾ ω) Fn ω)
1210, 11mpbir 230 . . . . . . . . . . . . 13 𝐻 Fn ω
13 fnfun 6602 . . . . . . . . . . . . 13 (𝐻 Fn ω → Fun 𝐻)
14 eluniima 7197 . . . . . . . . . . . . 13 (Fun 𝐻 → (𝑧 (𝐻 “ ω) ↔ ∃𝑣 ∈ ω 𝑧 ∈ (𝐻𝑣)))
1512, 13, 14mp2b 10 . . . . . . . . . . . 12 (𝑧 (𝐻 “ ω) ↔ ∃𝑣 ∈ ω 𝑧 ∈ (𝐻𝑣))
16 alephsson 10036 . . . . . . . . . . . . . . . 16 ran ℵ ⊆ On
171alephfplem3 10042 . . . . . . . . . . . . . . . 16 (𝑣 ∈ ω → (𝐻𝑣) ∈ ran ℵ)
1816, 17sselid 3942 . . . . . . . . . . . . . . 15 (𝑣 ∈ ω → (𝐻𝑣) ∈ On)
19 alephord2i 10013 . . . . . . . . . . . . . . 15 ((𝐻𝑣) ∈ On → (𝑧 ∈ (𝐻𝑣) → (ℵ‘𝑧) ∈ (ℵ‘(𝐻𝑣))))
2018, 19syl 17 . . . . . . . . . . . . . 14 (𝑣 ∈ ω → (𝑧 ∈ (𝐻𝑣) → (ℵ‘𝑧) ∈ (ℵ‘(𝐻𝑣))))
211alephfplem2 10041 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ ω → (𝐻‘suc 𝑣) = (ℵ‘(𝐻𝑣)))
22 peano2 7827 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ ω → suc 𝑣 ∈ ω)
23 fnfvelrn 7031 . . . . . . . . . . . . . . . . . . . 20 ((𝐻 Fn ω ∧ suc 𝑣 ∈ ω) → (𝐻‘suc 𝑣) ∈ ran 𝐻)
2412, 23mpan 688 . . . . . . . . . . . . . . . . . . 19 (suc 𝑣 ∈ ω → (𝐻‘suc 𝑣) ∈ ran 𝐻)
25 fnima 6631 . . . . . . . . . . . . . . . . . . . 20 (𝐻 Fn ω → (𝐻 “ ω) = ran 𝐻)
2612, 25ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝐻 “ ω) = ran 𝐻
2724, 26eleqtrrdi 2849 . . . . . . . . . . . . . . . . . 18 (suc 𝑣 ∈ ω → (𝐻‘suc 𝑣) ∈ (𝐻 “ ω))
2822, 27syl 17 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ ω → (𝐻‘suc 𝑣) ∈ (𝐻 “ ω))
2921, 28eqeltrrd 2839 . . . . . . . . . . . . . . . 16 (𝑣 ∈ ω → (ℵ‘(𝐻𝑣)) ∈ (𝐻 “ ω))
30 elssuni 4898 . . . . . . . . . . . . . . . 16 ((ℵ‘(𝐻𝑣)) ∈ (𝐻 “ ω) → (ℵ‘(𝐻𝑣)) ⊆ (𝐻 “ ω))
3129, 30syl 17 . . . . . . . . . . . . . . 15 (𝑣 ∈ ω → (ℵ‘(𝐻𝑣)) ⊆ (𝐻 “ ω))
3231sseld 3943 . . . . . . . . . . . . . 14 (𝑣 ∈ ω → ((ℵ‘𝑧) ∈ (ℵ‘(𝐻𝑣)) → (ℵ‘𝑧) ∈ (𝐻 “ ω)))
3320, 32syld 47 . . . . . . . . . . . . 13 (𝑣 ∈ ω → (𝑧 ∈ (𝐻𝑣) → (ℵ‘𝑧) ∈ (𝐻 “ ω)))
3433rexlimiv 3145 . . . . . . . . . . . 12 (∃𝑣 ∈ ω 𝑧 ∈ (𝐻𝑣) → (ℵ‘𝑧) ∈ (𝐻 “ ω))
3515, 34sylbi 216 . . . . . . . . . . 11 (𝑧 (𝐻 “ ω) → (ℵ‘𝑧) ∈ (𝐻 “ ω))
36 eleq2 2826 . . . . . . . . . . . 12 ( (𝐻 “ ω) = (ℵ‘𝑧) → (𝑧 (𝐻 “ ω) ↔ 𝑧 ∈ (ℵ‘𝑧)))
37 eleq2 2826 . . . . . . . . . . . 12 ( (𝐻 “ ω) = (ℵ‘𝑧) → ((ℵ‘𝑧) ∈ (𝐻 “ ω) ↔ (ℵ‘𝑧) ∈ (ℵ‘𝑧)))
3836, 37imbi12d 344 . . . . . . . . . . 11 ( (𝐻 “ ω) = (ℵ‘𝑧) → ((𝑧 (𝐻 “ ω) → (ℵ‘𝑧) ∈ (𝐻 “ ω)) ↔ (𝑧 ∈ (ℵ‘𝑧) → (ℵ‘𝑧) ∈ (ℵ‘𝑧))))
3935, 38mpbii 232 . . . . . . . . . 10 ( (𝐻 “ ω) = (ℵ‘𝑧) → (𝑧 ∈ (ℵ‘𝑧) → (ℵ‘𝑧) ∈ (ℵ‘𝑧)))
409, 39mtoi 198 . . . . . . . . 9 ( (𝐻 “ ω) = (ℵ‘𝑧) → ¬ 𝑧 ∈ (ℵ‘𝑧))
417, 40anim12i 613 . . . . . . . 8 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧)))
42 eloni 6327 . . . . . . . . . 10 (𝑧 ∈ On → Ord 𝑧)
438onordi 6428 . . . . . . . . . 10 Ord (ℵ‘𝑧)
44 ordtri4 6354 . . . . . . . . . 10 ((Ord 𝑧 ∧ Ord (ℵ‘𝑧)) → (𝑧 = (ℵ‘𝑧) ↔ (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧))))
4542, 43, 44sylancl 586 . . . . . . . . 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 256 . . . . . . 7 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → 𝑧 = (ℵ‘𝑧))
48 eqeq2 2748 . . . . . . . 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 256 . . . . . 6 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → 𝑧 = (𝐻 “ ω))
5150eqcomd 2742 . . . . 5 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → (𝐻 “ ω) = 𝑧)
5251fveq2d 6846 . . . 4 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → (ℵ‘ (𝐻 “ ω)) = (ℵ‘𝑧))
53 eqeq2 2748 . . . . 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 256 . . 3 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → (ℵ‘ (𝐻 “ ω)) = (𝐻 “ ω))
5655rexlimiva 3144 . 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 205  wa 396   = wceq 1541  wcel 2106  wrex 3073  wss 3910   cuni 4865  ran crn 5634  cres 5635  cima 5636  Ord word 6316  Oncon0 6317  suc csuc 6319  Fun wfun 6490   Fn wfn 6491  cfv 6496  ωcom 7802  reccrdg 8355  cardccrd 9871  cale 9872
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-oi 9446  df-har 9493  df-card 9875  df-aleph 9876
This theorem is referenced by:  alephfp2  10045
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