Theorem alephfp 9864

Description: The aleph function has a fixed point. Similar to Proposition 11.18 of [TakeutiZaring] p. 104, except that we construct an actual example of a fixed point rather than just showing its existence. See alephfp2 9865 for an abbreviated version just showing existence. (Contributed by NM, 6-Nov-2004.) (Proof shortened by Mario Carneiro, 15-May-2015.)

Hypothesis

Ref	Expression
alephfplem.1	⊢ 𝐻 = (rec(ℵ, ω) ↾ ω)

Assertion

Ref	Expression
alephfp	⊢ (ℵ‘∪ (𝐻 “ ω)) = ∪ (𝐻 “ ω)

Proof of Theorem alephfp

Dummy variables 𝑧 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		alephfplem.1	. . 3 ⊢ 𝐻 = (rec(ℵ, ω) ↾ ω)
2	1	alephfplem4 9863	. 2 ⊢ ∪ (𝐻 “ ω) ∈ ran ℵ
3		isinfcard 9848	. . 3 ⊢ ((ω ⊆ ∪ (𝐻 “ ω) ∧ (card‘∪ (𝐻 “ ω)) = ∪ (𝐻 “ ω)) ↔ ∪ (𝐻 “ ω) ∈ ran ℵ)
4		cardalephex 9846	. . . 4 ⊢ (ω ⊆ ∪ (𝐻 “ ω) → ((card‘∪ (𝐻 “ ω)) = ∪ (𝐻 “ ω) ↔ ∃𝑧 ∈ On ∪ (𝐻 “ ω) = (ℵ‘𝑧)))
5	4	biimpa 477	. . 3 ⊢ ((ω ⊆ ∪ (𝐻 “ ω) ∧ (card‘∪ (𝐻 “ ω)) = ∪ (𝐻 “ ω)) → ∃𝑧 ∈ On ∪ (𝐻 “ ω) = (ℵ‘𝑧))
6	3, 5	sylbir 234	. 2 ⊢ (∪ (𝐻 “ ω) ∈ ran ℵ → ∃𝑧 ∈ On ∪ (𝐻 “ ω) = (ℵ‘𝑧))
7		alephle 9844	. . . . . . . . 9 ⊢ (𝑧 ∈ On → 𝑧 ⊆ (ℵ‘𝑧))
8		alephon 9825	. . . . . . . . . . 11 ⊢ (ℵ‘𝑧) ∈ On
9	8	onirri 6373	. . . . . . . . . 10 ⊢ ¬ (ℵ‘𝑧) ∈ (ℵ‘𝑧)
10		frfnom 8266	. . . . . . . . . . . . . 14 ⊢ (rec(ℵ, ω) ↾ ω) Fn ω
11	1	fneq1i 6530	. . . . . . . . . . . . . 14 ⊢ (𝐻 Fn ω ↔ (rec(ℵ, ω) ↾ ω) Fn ω)
12	10, 11	mpbir 230	. . . . . . . . . . . . 13 ⊢ 𝐻 Fn ω
13		fnfun 6533	. . . . . . . . . . . . 13 ⊢ (𝐻 Fn ω → Fun 𝐻)
14		eluniima 7123	. . . . . . . . . . . . 13 ⊢ (Fun 𝐻 → (𝑧 ∈ ∪ (𝐻 “ ω) ↔ ∃𝑣 ∈ ω 𝑧 ∈ (𝐻‘𝑣)))
15	12, 13, 14	mp2b 10	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ∪ (𝐻 “ ω) ↔ ∃𝑣 ∈ ω 𝑧 ∈ (𝐻‘𝑣))
16		alephsson 9856	. . . . . . . . . . . . . . . 16 ⊢ ran ℵ ⊆ On
17	1	alephfplem3 9862	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ ω → (𝐻‘𝑣) ∈ ran ℵ)
18	16, 17	sselid 3919	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ ω → (𝐻‘𝑣) ∈ On)
19		alephord2i 9833	. . . . . . . . . . . . . . 15 ⊢ ((𝐻‘𝑣) ∈ On → (𝑧 ∈ (𝐻‘𝑣) → (ℵ‘𝑧) ∈ (ℵ‘(𝐻‘𝑣))))
20	18, 19	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ ω → (𝑧 ∈ (𝐻‘𝑣) → (ℵ‘𝑧) ∈ (ℵ‘(𝐻‘𝑣))))
21	1	alephfplem2 9861	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ ω → (𝐻‘suc 𝑣) = (ℵ‘(𝐻‘𝑣)))
22		peano2 7737	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ∈ ω → suc 𝑣 ∈ ω)
23		fnfvelrn 6958	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐻 Fn ω ∧ suc 𝑣 ∈ ω) → (𝐻‘suc 𝑣) ∈ ran 𝐻)
24	12, 23	mpan 687	. . . . . . . . . . . . . . . . . . 19 ⊢ (suc 𝑣 ∈ ω → (𝐻‘suc 𝑣) ∈ ran 𝐻)
25		fnima 6563	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐻 Fn ω → (𝐻 “ ω) = ran 𝐻)
26	12, 25	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐻 “ ω) = ran 𝐻
27	24, 26	eleqtrrdi 2850	. . . . . . . . . . . . . . . . . 18 ⊢ (suc 𝑣 ∈ ω → (𝐻‘suc 𝑣) ∈ (𝐻 “ ω))
28	22, 27	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ ω → (𝐻‘suc 𝑣) ∈ (𝐻 “ ω))
29	21, 28	eqeltrrd 2840	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ ω → (ℵ‘(𝐻‘𝑣)) ∈ (𝐻 “ ω))
30		elssuni 4871	. . . . . . . . . . . . . . . 16 ⊢ ((ℵ‘(𝐻‘𝑣)) ∈ (𝐻 “ ω) → (ℵ‘(𝐻‘𝑣)) ⊆ ∪ (𝐻 “ ω))
31	29, 30	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ ω → (ℵ‘(𝐻‘𝑣)) ⊆ ∪ (𝐻 “ ω))
32	31	sseld 3920	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ ω → ((ℵ‘𝑧) ∈ (ℵ‘(𝐻‘𝑣)) → (ℵ‘𝑧) ∈ ∪ (𝐻 “ ω)))
33	20, 32	syld 47	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ ω → (𝑧 ∈ (𝐻‘𝑣) → (ℵ‘𝑧) ∈ ∪ (𝐻 “ ω)))
34	33	rexlimiv 3209	. . . . . . . . . . . 12 ⊢ (∃𝑣 ∈ ω 𝑧 ∈ (𝐻‘𝑣) → (ℵ‘𝑧) ∈ ∪ (𝐻 “ ω))
35	15, 34	sylbi 216	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ∪ (𝐻 “ ω) → (ℵ‘𝑧) ∈ ∪ (𝐻 “ ω))
36		eleq2 2827	. . . . . . . . . . . 12 ⊢ (∪ (𝐻 “ ω) = (ℵ‘𝑧) → (𝑧 ∈ ∪ (𝐻 “ ω) ↔ 𝑧 ∈ (ℵ‘𝑧)))
37		eleq2 2827	. . . . . . . . . . . 12 ⊢ (∪ (𝐻 “ ω) = (ℵ‘𝑧) → ((ℵ‘𝑧) ∈ ∪ (𝐻 “ ω) ↔ (ℵ‘𝑧) ∈ (ℵ‘𝑧)))
38	36, 37	imbi12d 345	. . . . . . . . . . 11 ⊢ (∪ (𝐻 “ ω) = (ℵ‘𝑧) → ((𝑧 ∈ ∪ (𝐻 “ ω) → (ℵ‘𝑧) ∈ ∪ (𝐻 “ ω)) ↔ (𝑧 ∈ (ℵ‘𝑧) → (ℵ‘𝑧) ∈ (ℵ‘𝑧))))
39	35, 38	mpbii 232	. . . . . . . . . 10 ⊢ (∪ (𝐻 “ ω) = (ℵ‘𝑧) → (𝑧 ∈ (ℵ‘𝑧) → (ℵ‘𝑧) ∈ (ℵ‘𝑧)))
40	9, 39	mtoi 198	. . . . . . . . 9 ⊢ (∪ (𝐻 “ ω) = (ℵ‘𝑧) → ¬ 𝑧 ∈ (ℵ‘𝑧))
41	7, 40	anim12i 613	. . . . . . . 8 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧)))
42		eloni 6276	. . . . . . . . . 10 ⊢ (𝑧 ∈ On → Ord 𝑧)
43	8	onordi 6371	. . . . . . . . . 10 ⊢ Ord (ℵ‘𝑧)
44		ordtri4 6303	. . . . . . . . . 10 ⊢ ((Ord 𝑧 ∧ Ord (ℵ‘𝑧)) → (𝑧 = (ℵ‘𝑧) ↔ (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧))))
45	42, 43, 44	sylancl 586	. . . . . . . . 9 ⊢ (𝑧 ∈ On → (𝑧 = (ℵ‘𝑧) ↔ (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧))))
46	45	adantr 481	. . . . . . . 8 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → (𝑧 = (ℵ‘𝑧) ↔ (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧))))
47	41, 46	mpbird 256	. . . . . . 7 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → 𝑧 = (ℵ‘𝑧))
48		eqeq2 2750	. . . . . . . 8 ⊢ (∪ (𝐻 “ ω) = (ℵ‘𝑧) → (𝑧 = ∪ (𝐻 “ ω) ↔ 𝑧 = (ℵ‘𝑧)))
49	48	adantl 482	. . . . . . 7 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → (𝑧 = ∪ (𝐻 “ ω) ↔ 𝑧 = (ℵ‘𝑧)))
50	47, 49	mpbird 256	. . . . . 6 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → 𝑧 = ∪ (𝐻 “ ω))
51	50	eqcomd 2744	. . . . 5 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → ∪ (𝐻 “ ω) = 𝑧)
52	51	fveq2d 6778	. . . 4 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → (ℵ‘∪ (𝐻 “ ω)) = (ℵ‘𝑧))
53		eqeq2 2750	. . . . 5 ⊢ (∪ (𝐻 “ ω) = (ℵ‘𝑧) → ((ℵ‘∪ (𝐻 “ ω)) = ∪ (𝐻 “ ω) ↔ (ℵ‘∪ (𝐻 “ ω)) = (ℵ‘𝑧)))
54	53	adantl 482	. . . 4 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → ((ℵ‘∪ (𝐻 “ ω)) = ∪ (𝐻 “ ω) ↔ (ℵ‘∪ (𝐻 “ ω)) = (ℵ‘𝑧)))
55	52, 54	mpbird 256	. . 3 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → (ℵ‘∪ (𝐻 “ ω)) = ∪ (𝐻 “ ω))
56	55	rexlimiva 3210	. 2 ⊢ (∃𝑧 ∈ On ∪ (𝐻 “ ω) = (ℵ‘𝑧) → (ℵ‘∪ (𝐻 “ ω)) = ∪ (𝐻 “ ω))
57	2, 6, 56	mp2b 10	1 ⊢ (ℵ‘∪ (𝐻 “ ω)) = ∪ (𝐻 “ ω)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃wrex 3065   ⊆ wss 3887  ∪ cuni 4839  ran crn 5590   ↾ cres 5591   “ cima 5592  Ord word 6265  Oncon0 6266  suc csuc 6268  Fun wfun 6427   Fn wfn 6428  ‘cfv 6433  ωcom 7712  reccrdg 8240  cardccrd 9693  ℵcale 9694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-oi 9269  df-har 9316  df-card 9697  df-aleph 9698

This theorem is referenced by:  alephfp2  9865