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Theorem alephfp2 9609
Description: The aleph function has at least one fixed point. Proposition 11.18 of [TakeutiZaring] p. 104. See alephfp 9608 for an actual example of a fixed point. Compare the inequality alephle 9588 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 9600 . . 3 ran ℵ ⊆ On
2 eqid 2738 . . . 4 (rec(ℵ, ω) ↾ ω) = (rec(ℵ, ω) ↾ ω)
32alephfplem4 9607 . . 3 ((rec(ℵ, ω) ↾ ω) “ ω) ∈ ran ℵ
41, 3sselii 3874 . 2 ((rec(ℵ, ω) ↾ ω) “ ω) ∈ On
52alephfp 9608 . 2 (ℵ‘ ((rec(ℵ, ω) ↾ ω) “ ω)) = ((rec(ℵ, ω) ↾ ω) “ ω)
6 fveq2 6674 . . . 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 2754 . . 3 (𝑥 = ((rec(ℵ, ω) ↾ ω) “ ω) → ((ℵ‘𝑥) = 𝑥 ↔ (ℵ‘ ((rec(ℵ, ω) ↾ ω) “ ω)) = ((rec(ℵ, ω) ↾ ω) “ ω)))
98rspcev 3526 . 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 692 1 𝑥 ∈ On (ℵ‘𝑥) = 𝑥
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2114  wrex 3054   cuni 4796  ran crn 5526  cres 5527  cima 5528  Oncon0 6172  cfv 6339  ωcom 7599  reccrdg 8074  cale 9438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-inf2 9177
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-se 5484  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7127  df-om 7600  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-er 8320  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-oi 9047  df-har 9094  df-card 9441  df-aleph 9442
This theorem is referenced by: (None)
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