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Theorem alephfp2 8815
 Description: The aleph function has at least one fixed point. Proposition 11.18 of [TakeutiZaring] p. 104. See alephfp 8814 for an actual example of a fixed point. Compare the inequality alephle 8794 that holds in general. Note that if 𝑥 is a fixed point, then ℵ‘ℵ‘ℵ‘... ℵ‘𝑥 = 𝑥. (Contributed by NM, 6-Nov-2004.) (Revised by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
alephfp2 𝑥 ∈ On (ℵ‘𝑥) = 𝑥

Proof of Theorem alephfp2
StepHypRef Expression
1 alephsson 8806 . . 3 ran ℵ ⊆ On
2 eqid 2610 . . . 4 (rec(ℵ, ω) ↾ ω) = (rec(ℵ, ω) ↾ ω)
32alephfplem4 8813 . . 3 ((rec(ℵ, ω) ↾ ω) “ ω) ∈ ran ℵ
41, 3sselii 3565 . 2 ((rec(ℵ, ω) ↾ ω) “ ω) ∈ On
52alephfp 8814 . 2 (ℵ‘ ((rec(ℵ, ω) ↾ ω) “ ω)) = ((rec(ℵ, ω) ↾ ω) “ ω)
6 fveq2 6103 . . . 4 (𝑥 = ((rec(ℵ, ω) ↾ ω) “ ω) → (ℵ‘𝑥) = (ℵ‘ ((rec(ℵ, ω) ↾ ω) “ ω)))
7 id 22 . . . 4 (𝑥 = ((rec(ℵ, ω) ↾ ω) “ ω) → 𝑥 = ((rec(ℵ, ω) ↾ ω) “ ω))
86, 7eqeq12d 2625 . . 3 (𝑥 = ((rec(ℵ, ω) ↾ ω) “ ω) → ((ℵ‘𝑥) = 𝑥 ↔ (ℵ‘ ((rec(ℵ, ω) ↾ ω) “ ω)) = ((rec(ℵ, ω) ↾ ω) “ ω)))
98rspcev 3282 . 2 (( ((rec(ℵ, ω) ↾ ω) “ ω) ∈ On ∧ (ℵ‘ ((rec(ℵ, ω) ↾ ω) “ ω)) = ((rec(ℵ, ω) ↾ ω) “ ω)) → ∃𝑥 ∈ On (ℵ‘𝑥) = 𝑥)
104, 5, 9mp2an 704 1 𝑥 ∈ On (ℵ‘𝑥) = 𝑥
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  ∪ cuni 4372  ran crn 5039   ↾ cres 5040   “ cima 5041  Oncon0 5640  ‘cfv 5804  ωcom 6957  reccrdg 7392  ℵcale 8645 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-har 8346  df-card 8648  df-aleph 8649 This theorem is referenced by: (None)
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