Theorem alephfp2 10100

Description: The aleph function has at least one fixed point. Proposition 11.18 of [TakeutiZaring] p. 104. See alephfp 10099 for an actual example of a fixed point. Compare the inequality alephle 10079 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 10091	. . 3 ⊢ ran ℵ ⊆ On
2		eqid 2762	. . . 4 ⊢ (rec(ℵ, ω) ↾ ω) = (rec(ℵ, ω) ↾ ω)
3	2	alephfplem4 10098	. . 3 ⊢ ∪ ((rec(ℵ, ω) ↾ ω) “ ω) ∈ ran ℵ
4	1, 3	sselii 3933	. 2 ⊢ ∪ ((rec(ℵ, ω) ↾ ω) “ ω) ∈ On
5	2	alephfp 10099	. 2 ⊢ (ℵ‘∪ ((rec(ℵ, ω) ↾ ω) “ ω)) = ∪ ((rec(ℵ, ω) ↾ ω) “ ω)
6		fveq2 6881	. . . 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 3580	. 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 704	1 ⊢ ∃𝑥 ∈ On (ℵ‘𝑥) = 𝑥

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2142  ∃wrex 3088  ∪ cuni 4871  ran crn 5661   ↾ cres 5662   “ cima 5663  Oncon0 6360  ‘cfv 6536  ωcom 7860  reccrdg 8394  ℵcale 9929

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-inf2 9608

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  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 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-se 5614  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7369  df-ov 7415  df-om 7861  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-er 8692  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-oi 9470  df-har 9517  df-card 9932  df-aleph 9933

This theorem is used by: (None)