Theorem alephfp2 10115

Description: The aleph function has at least one fixed point. Proposition 11.18 of [TakeutiZaring] p. 104. See alephfp 10114 for an actual example of a fixed point. Compare the inequality alephle 10094 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 10106	. . 3 ⊢ ran ℵ ⊆ On
2		eqid 2762	. . . 4 ⊢ (rec(ℵ, ω) ↾ ω) = (rec(ℵ, ω) ↾ ω)
3	2	alephfplem4 10113	. . 3 ⊢ ∪ ((rec(ℵ, ω) ↾ ω) “ ω) ∈ ran ℵ
4	1, 3	sselii 3931	. 2 ⊢ ∪ ((rec(ℵ, ω) ↾ ω) “ ω) ∈ On
5	2	alephfp 10114	. 2 ⊢ (ℵ‘∪ ((rec(ℵ, ω) ↾ ω) “ ω)) = ∪ ((rec(ℵ, ω) ↾ ω) “ ω)
6		fveq2 6882	. . . 4 ⊢ (𝑥 = ∪ ((rec(ℵ, ω) ↾ ω) “ ω) → (ℵ‘𝑥) = (ℵ‘∪ ((rec(ℵ, ω) ↾ ω) “ ω)))
7		id 23	. . . 4 ⊢ (𝑥 = ∪ ((rec(ℵ, ω) ↾ ω) “ ω) → 𝑥 = ∪ ((rec(ℵ, ω) ↾ ω) “ ω))
8	6, 7	eqeq12d 2778	. . 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 705	1 ⊢ ∃𝑥 ∈ On (ℵ‘𝑥) = 𝑥

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  ∃wrex 3088  ∪ cuni 4870  ran crn 5660   ↾ cres 5661   “ cima 5662  Oncon0 6361  ‘cfv 6537  ωcom 7865  reccrdg 8401  ℵcale 9944

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-inf2 9623

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7373  df-ov 7419  df-om 7866  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-er 8699  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-oi 9485  df-har 9532  df-card 9947  df-aleph 9948

This theorem is used by: (None)