Theorem alephfp2 10022

Description: The aleph function has at least one fixed point. Proposition 11.18 of [TakeutiZaring] p. 104. See alephfp 10021 for an actual example of a fixed point. Compare the inequality alephle 10001 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

Step	Hyp	Ref	Expression
1		alephsson 10013	. . 3 ⊢ ran ℵ ⊆ On
2		eqid 2729	. . . 4 ⊢ (rec(ℵ, ω) ↾ ω) = (rec(ℵ, ω) ↾ ω)
3	2	alephfplem4 10020	. . 3 ⊢ ∪ ((rec(ℵ, ω) ↾ ω) “ ω) ∈ ran ℵ
4	1, 3	sselii 3934	. 2 ⊢ ∪ ((rec(ℵ, ω) ↾ ω) “ ω) ∈ On
5	2	alephfp 10021	. 2 ⊢ (ℵ‘∪ ((rec(ℵ, ω) ↾ ω) “ ω)) = ∪ ((rec(ℵ, ω) ↾ ω) “ ω)
6		fveq2 6826	. . . 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))
8	6, 7	eqeq12d 2745	. . 3 ⊢ (𝑥 = ∪ ((rec(ℵ, ω) ↾ ω) “ ω) → ((ℵ‘𝑥) = 𝑥 ↔ (ℵ‘∪ ((rec(ℵ, ω) ↾ ω) “ ω)) = ∪ ((rec(ℵ, ω) ↾ ω) “ ω)))
9	8	rspcev 3579	. 2 ⊢ ((∪ ((rec(ℵ, ω) ↾ ω) “ ω) ∈ On ∧ (ℵ‘∪ ((rec(ℵ, ω) ↾ ω) “ ω)) = ∪ ((rec(ℵ, ω) ↾ ω) “ ω)) → ∃𝑥 ∈ On (ℵ‘𝑥) = 𝑥)
10	4, 5, 9	mp2an 692	1 ⊢ ∃𝑥 ∈ On (ℵ‘𝑥) = 𝑥

Syntax hints:   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  ∪ cuni 4861  ran crn 5624   ↾ cres 5625   “ cima 5626  Oncon0 6311  ‘cfv 6486  ωcom 7806  reccrdg 8338  ℵcale 9851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-inf2 9556

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7310  df-ov 7356  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-oi 9421  df-har 9468  df-card 9854  df-aleph 9855

This theorem is referenced by: (None)