Theorem alephfp2 9796

Description: The aleph function has at least one fixed point. Proposition 11.18 of [TakeutiZaring] p. 104. See alephfp 9795 for an actual example of a fixed point. Compare the inequality alephle 9775 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 9787	. . 3 ⊢ ran ℵ ⊆ On
2		eqid 2738	. . . 4 ⊢ (rec(ℵ, ω) ↾ ω) = (rec(ℵ, ω) ↾ ω)
3	2	alephfplem4 9794	. . 3 ⊢ ∪ ((rec(ℵ, ω) ↾ ω) “ ω) ∈ ran ℵ
4	1, 3	sselii 3914	. 2 ⊢ ∪ ((rec(ℵ, ω) ↾ ω) “ ω) ∈ On
5	2	alephfp 9795	. 2 ⊢ (ℵ‘∪ ((rec(ℵ, ω) ↾ ω) “ ω)) = ∪ ((rec(ℵ, ω) ↾ ω) “ ω)
6		fveq2 6756	. . . 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 2754	. . 3 ⊢ (𝑥 = ∪ ((rec(ℵ, ω) ↾ ω) “ ω) → ((ℵ‘𝑥) = 𝑥 ↔ (ℵ‘∪ ((rec(ℵ, ω) ↾ ω) “ ω)) = ∪ ((rec(ℵ, ω) ↾ ω) “ ω)))
9	8	rspcev 3552	. 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 688	1 ⊢ ∃𝑥 ∈ On (ℵ‘𝑥) = 𝑥

Syntax hints:   = wceq 1539   ∈ wcel 2108  ∃wrex 3064  ∪ cuni 4836  ran crn 5581   ↾ cres 5582   “ cima 5583  Oncon0 6251  ‘cfv 6418  ωcom 7687  reccrdg 8211  ℵcale 9625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-oi 9199  df-har 9246  df-card 9628  df-aleph 9629

This theorem is referenced by: (None)