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Theorem alephfp2 9131
Description: The aleph function has at least one fixed point. Proposition 11.18 of [TakeutiZaring] p. 104. See alephfp 9130 for an actual example of a fixed point. Compare the inequality alephle 9110 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 9122 . . 3 ran ℵ ⊆ On
2 eqid 2770 . . . 4 (rec(ℵ, ω) ↾ ω) = (rec(ℵ, ω) ↾ ω)
32alephfplem4 9129 . . 3 ((rec(ℵ, ω) ↾ ω) “ ω) ∈ ran ℵ
41, 3sselii 3747 . 2 ((rec(ℵ, ω) ↾ ω) “ ω) ∈ On
52alephfp 9130 . 2 (ℵ‘ ((rec(ℵ, ω) ↾ ω) “ ω)) = ((rec(ℵ, ω) ↾ ω) “ ω)
6 fveq2 6332 . . . 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 2785 . . 3 (𝑥 = ((rec(ℵ, ω) ↾ ω) “ ω) → ((ℵ‘𝑥) = 𝑥 ↔ (ℵ‘ ((rec(ℵ, ω) ↾ ω) “ ω)) = ((rec(ℵ, ω) ↾ ω) “ ω)))
98rspcev 3458 . 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 664 1 𝑥 ∈ On (ℵ‘𝑥) = 𝑥
Colors of variables: wff setvar class
Syntax hints:   = wceq 1630  wcel 2144  wrex 3061   cuni 4572  ran crn 5250  cres 5251  cima 5252  Oncon0 5866  cfv 6031  ωcom 7211  reccrdg 7657  cale 8961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-inf2 8701
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  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 6753  df-om 7212  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-oi 8570  df-har 8618  df-card 8964  df-aleph 8965
This theorem is referenced by: (None)
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