Theorem alephfp 10091

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 10092 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 10090	. 2 ⊢ ∪ (𝐻 “ ω) ∈ ran ℵ
3		isinfcard 10075	. . 3 ⊢ ((ω ⊆ ∪ (𝐻 “ ω) ∧ (card‘∪ (𝐻 “ ω)) = ∪ (𝐻 “ ω)) ↔ ∪ (𝐻 “ ω) ∈ ran ℵ)
4		cardalephex 10073	. . . 4 ⊢ (ω ⊆ ∪ (𝐻 “ ω) → ((card‘∪ (𝐻 “ ω)) = ∪ (𝐻 “ ω) ↔ ∃𝑧 ∈ On ∪ (𝐻 “ ω) = (ℵ‘𝑧)))
5	4	biimpa 481	. . 3 ⊢ ((ω ⊆ ∪ (𝐻 “ ω) ∧ (card‘∪ (𝐻 “ ω)) = ∪ (𝐻 “ ω)) → ∃𝑧 ∈ On ∪ (𝐻 “ ω) = (ℵ‘𝑧))
6	3, 5	sylbir 238	. 2 ⊢ (∪ (𝐻 “ ω) ∈ ran ℵ → ∃𝑧 ∈ On ∪ (𝐻 “ ω) = (ℵ‘𝑧))
7		alephle 10071	. . . . . . . . 9 ⊢ (𝑧 ∈ On → 𝑧 ⊆ (ℵ‘𝑧))
8		alephon 10052	. . . . . . . . . . 11 ⊢ (ℵ‘𝑧) ∈ On
9	8	onirri 6475	. . . . . . . . . 10 ⊢ ¬ (ℵ‘𝑧) ∈ (ℵ‘𝑧)
10		frfnom 8421	. . . . . . . . . . . . . 14 ⊢ (rec(ℵ, ω) ↾ ω) Fn ω
11	1	fneq1i 6632	. . . . . . . . . . . . . 14 ⊢ (𝐻 Fn ω ↔ (rec(ℵ, ω) ↾ ω) Fn ω)
12	10, 11	mpbir 234	. . . . . . . . . . . . 13 ⊢ 𝐻 Fn ω
13		fnfun 6635	. . . . . . . . . . . . 13 ⊢ (𝐻 Fn ω → Fun 𝐻)
14		eluniima 7248	. . . . . . . . . . . . 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 10083	. . . . . . . . . . . . . . . 16 ⊢ ran ℵ ⊆ On
17	1	alephfplem3 10089	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ ω → (𝐻‘𝑣) ∈ ran ℵ)
18	16, 17	sselid 3934	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ ω → (𝐻‘𝑣) ∈ On)
19		alephord2i 10060	. . . . . . . . . . . . . . 15 ⊢ ((𝐻‘𝑣) ∈ On → (𝑧 ∈ (𝐻‘𝑣) → (ℵ‘𝑧) ∈ (ℵ‘(𝐻‘𝑣))))
20	18, 19	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ ω → (𝑧 ∈ (𝐻‘𝑣) → (ℵ‘𝑧) ∈ (ℵ‘(𝐻‘𝑣))))
21	1	alephfplem2 10088	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ ω → (𝐻‘suc 𝑣) = (ℵ‘(𝐻‘𝑣)))
22		peano2 7885	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ∈ ω → suc 𝑣 ∈ ω)
23		fnfvelrn 7075	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐻 Fn ω ∧ suc 𝑣 ∈ ω) → (𝐻‘suc 𝑣) ∈ ran 𝐻)
24	12, 23	mpan 702	. . . . . . . . . . . . . . . . . . 19 ⊢ (suc 𝑣 ∈ ω → (𝐻‘suc 𝑣) ∈ ran 𝐻)
25		fnima 6665	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐻 Fn ω → (𝐻 “ ω) = ran 𝐻)
26	12, 25	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐻 “ ω) = ran 𝐻
27	24, 26	eleqtrrdi 2872	. . . . . . . . . . . . . . . . . 18 ⊢ (suc 𝑣 ∈ ω → (𝐻‘suc 𝑣) ∈ (𝐻 “ ω))
28	22, 27	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ ω → (𝐻‘suc 𝑣) ∈ (𝐻 “ ω))
29	21, 28	eqeltrrd 2862	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ ω → (ℵ‘(𝐻‘𝑣)) ∈ (𝐻 “ ω))
30		elssuni 4903	. . . . . . . . . . . . . . . 16 ⊢ ((ℵ‘(𝐻‘𝑣)) ∈ (𝐻 “ ω) → (ℵ‘(𝐻‘𝑣)) ⊆ ∪ (𝐻 “ ω))
31	29, 30	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ ω → (ℵ‘(𝐻‘𝑣)) ⊆ ∪ (𝐻 “ ω))
32	31	sseld 3935	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ ω → ((ℵ‘𝑧) ∈ (ℵ‘(𝐻‘𝑣)) → (ℵ‘𝑧) ∈ ∪ (𝐻 “ ω)))
33	20, 32	syld 48	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ ω → (𝑧 ∈ (𝐻‘𝑣) → (ℵ‘𝑧) ∈ ∪ (𝐻 “ ω)))
34	33	rexlimiv 3157	. . . . . . . . . . . 12 ⊢ (∃𝑣 ∈ ω 𝑧 ∈ (𝐻‘𝑣) → (ℵ‘𝑧) ∈ ∪ (𝐻 “ ω))
35	15, 34	sylbi 220	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ∪ (𝐻 “ ω) → (ℵ‘𝑧) ∈ ∪ (𝐻 “ ω))
36		eleq2 2850	. . . . . . . . . . . 12 ⊢ (∪ (𝐻 “ ω) = (ℵ‘𝑧) → (𝑧 ∈ ∪ (𝐻 “ ω) ↔ 𝑧 ∈ (ℵ‘𝑧)))
37		eleq2 2850	. . . . . . . . . . . 12 ⊢ (∪ (𝐻 “ ω) = (ℵ‘𝑧) → ((ℵ‘𝑧) ∈ ∪ (𝐻 “ ω) ↔ (ℵ‘𝑧) ∈ (ℵ‘𝑧)))
38	36, 37	imbi12d 347	. . . . . . . . . . 11 ⊢ (∪ (𝐻 “ ω) = (ℵ‘𝑧) → ((𝑧 ∈ ∪ (𝐻 “ ω) → (ℵ‘𝑧) ∈ ∪ (𝐻 “ ω)) ↔ (𝑧 ∈ (ℵ‘𝑧) → (ℵ‘𝑧) ∈ (ℵ‘𝑧))))
39	35, 38	mpbii 236	. . . . . . . . . 10 ⊢ (∪ (𝐻 “ ω) = (ℵ‘𝑧) → (𝑧 ∈ (ℵ‘𝑧) → (ℵ‘𝑧) ∈ (ℵ‘𝑧)))
40	9, 39	mtoi 202	. . . . . . . . 9 ⊢ (∪ (𝐻 “ ω) = (ℵ‘𝑧) → ¬ 𝑧 ∈ (ℵ‘𝑧))
41	7, 40	anim12i 624	. . . . . . . 8 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧)))
42		eloni 6370	. . . . . . . . . 10 ⊢ (𝑧 ∈ On → Ord 𝑧)
43	8	onordi 6474	. . . . . . . . . 10 ⊢ Ord (ℵ‘𝑧)
44		ordtri4 6398	. . . . . . . . . 10 ⊢ ((Ord 𝑧 ∧ Ord (ℵ‘𝑧)) → (𝑧 = (ℵ‘𝑧) ↔ (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧))))
45	42, 43, 44	sylancl 597	. . . . . . . . 9 ⊢ (𝑧 ∈ On → (𝑧 = (ℵ‘𝑧) ↔ (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧))))
46	45	adantr 485	. . . . . . . 8 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → (𝑧 = (ℵ‘𝑧) ↔ (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧))))
47	41, 46	mpbird 260	. . . . . . 7 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → 𝑧 = (ℵ‘𝑧))
48		eqeq2 2773	. . . . . . . 8 ⊢ (∪ (𝐻 “ ω) = (ℵ‘𝑧) → (𝑧 = ∪ (𝐻 “ ω) ↔ 𝑧 = (ℵ‘𝑧)))
49	48	adantl 486	. . . . . . 7 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → (𝑧 = ∪ (𝐻 “ ω) ↔ 𝑧 = (ℵ‘𝑧)))
50	47, 49	mpbird 260	. . . . . 6 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → 𝑧 = ∪ (𝐻 “ ω))
51	50	eqcomd 2767	. . . . 5 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → ∪ (𝐻 “ ω) = 𝑧)
52	51	fveq2d 6885	. . . 4 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → (ℵ‘∪ (𝐻 “ ω)) = (ℵ‘𝑧))
53		eqeq2 2773	. . . . 5 ⊢ (∪ (𝐻 “ ω) = (ℵ‘𝑧) → ((ℵ‘∪ (𝐻 “ ω)) = ∪ (𝐻 “ ω) ↔ (ℵ‘∪ (𝐻 “ ω)) = (ℵ‘𝑧)))
54	53	adantl 486	. . . 4 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → ((ℵ‘∪ (𝐻 “ ω)) = ∪ (𝐻 “ ω) ↔ (ℵ‘∪ (𝐻 “ ω)) = (ℵ‘𝑧)))
55	52, 54	mpbird 260	. . 3 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → (ℵ‘∪ (𝐻 “ ω)) = ∪ (𝐻 “ ω))
56	55	rexlimiva 3156	. 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 209   ∧ wa 400   = wceq 1568   ∈ wcel 2141  ∃wrex 3087   ⊆ wss 3904  ∪ cuni 4871  ran crn 5662   ↾ cres 5663   “ cima 5664  Ord word 6359  Oncon0 6360  suc csuc 6362  Fun wfun 6530   Fn wfn 6531  ‘cfv 6536  ωcom 7861  reccrdg 8395  cardccrd 9920  ℵcale 9921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9609

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  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 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  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 7367  df-ov 7413  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-er 8693  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-oi 9471  df-har 9518  df-card 9924  df-aleph 9925

This theorem is referenced by:  alephfp2  10092