Theorem alephfp 10019

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 10020 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 10018	. 2 ⊢ ∪ (𝐻 “ ω) ∈ ran ℵ
3		isinfcard 10003	. . 3 ⊢ ((ω ⊆ ∪ (𝐻 “ ω) ∧ (card‘∪ (𝐻 “ ω)) = ∪ (𝐻 “ ω)) ↔ ∪ (𝐻 “ ω) ∈ ran ℵ)
4		cardalephex 10001	. . . 4 ⊢ (ω ⊆ ∪ (𝐻 “ ω) → ((card‘∪ (𝐻 “ ω)) = ∪ (𝐻 “ ω) ↔ ∃𝑧 ∈ On ∪ (𝐻 “ ω) = (ℵ‘𝑧)))
5	4	biimpa 476	. . 3 ⊢ ((ω ⊆ ∪ (𝐻 “ ω) ∧ (card‘∪ (𝐻 “ ω)) = ∪ (𝐻 “ ω)) → ∃𝑧 ∈ On ∪ (𝐻 “ ω) = (ℵ‘𝑧))
6	3, 5	sylbir 235	. 2 ⊢ (∪ (𝐻 “ ω) ∈ ran ℵ → ∃𝑧 ∈ On ∪ (𝐻 “ ω) = (ℵ‘𝑧))
7		alephle 9999	. . . . . . . . 9 ⊢ (𝑧 ∈ On → 𝑧 ⊆ (ℵ‘𝑧))
8		alephon 9980	. . . . . . . . . . 11 ⊢ (ℵ‘𝑧) ∈ On
9	8	onirri 6426	. . . . . . . . . 10 ⊢ ¬ (ℵ‘𝑧) ∈ (ℵ‘𝑧)
10		frfnom 8363	. . . . . . . . . . . . . 14 ⊢ (rec(ℵ, ω) ↾ ω) Fn ω
11	1	fneq1i 6584	. . . . . . . . . . . . . 14 ⊢ (𝐻 Fn ω ↔ (rec(ℵ, ω) ↾ ω) Fn ω)
12	10, 11	mpbir 231	. . . . . . . . . . . . 13 ⊢ 𝐻 Fn ω
13		fnfun 6587	. . . . . . . . . . . . 13 ⊢ (𝐻 Fn ω → Fun 𝐻)
14		eluniima 7194	. . . . . . . . . . . . 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 10011	. . . . . . . . . . . . . . . 16 ⊢ ran ℵ ⊆ On
17	1	alephfplem3 10017	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ ω → (𝐻‘𝑣) ∈ ran ℵ)
18	16, 17	sselid 3915	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ ω → (𝐻‘𝑣) ∈ On)
19		alephord2i 9988	. . . . . . . . . . . . . . 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 10016	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ ω → (𝐻‘suc 𝑣) = (ℵ‘(𝐻‘𝑣)))
22		peano2 7830	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ∈ ω → suc 𝑣 ∈ ω)
23		fnfvelrn 7021	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐻 Fn ω ∧ suc 𝑣 ∈ ω) → (𝐻‘suc 𝑣) ∈ ran 𝐻)
24	12, 23	mpan 691	. . . . . . . . . . . . . . . . . . 19 ⊢ (suc 𝑣 ∈ ω → (𝐻‘suc 𝑣) ∈ ran 𝐻)
25		fnima 6617	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐻 Fn ω → (𝐻 “ ω) = ran 𝐻)
26	12, 25	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐻 “ ω) = ran 𝐻
27	24, 26	eleqtrrdi 2846	. . . . . . . . . . . . . . . . . 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 2836	. . . . . . . . . . . . . . . 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 3916	. . . . . . . . . . . . . 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 3129	. . . . . . . . . . . 12 ⊢ (∃𝑣 ∈ ω 𝑧 ∈ (𝐻‘𝑣) → (ℵ‘𝑧) ∈ ∪ (𝐻 “ ω))
35	15, 34	sylbi 217	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ∪ (𝐻 “ ω) → (ℵ‘𝑧) ∈ ∪ (𝐻 “ ω))
36		eleq2 2824	. . . . . . . . . . . 12 ⊢ (∪ (𝐻 “ ω) = (ℵ‘𝑧) → (𝑧 ∈ ∪ (𝐻 “ ω) ↔ 𝑧 ∈ (ℵ‘𝑧)))
37		eleq2 2824	. . . . . . . . . . . 12 ⊢ (∪ (𝐻 “ ω) = (ℵ‘𝑧) → ((ℵ‘𝑧) ∈ ∪ (𝐻 “ ω) ↔ (ℵ‘𝑧) ∈ (ℵ‘𝑧)))
38	36, 37	imbi12d 344	. . . . . . . . . . 11 ⊢ (∪ (𝐻 “ ω) = (ℵ‘𝑧) → ((𝑧 ∈ ∪ (𝐻 “ ω) → (ℵ‘𝑧) ∈ ∪ (𝐻 “ ω)) ↔ (𝑧 ∈ (ℵ‘𝑧) → (ℵ‘𝑧) ∈ (ℵ‘𝑧))))
39	35, 38	mpbii 233	. . . . . . . . . 10 ⊢ (∪ (𝐻 “ ω) = (ℵ‘𝑧) → (𝑧 ∈ (ℵ‘𝑧) → (ℵ‘𝑧) ∈ (ℵ‘𝑧)))
40	9, 39	mtoi 199	. . . . . . . . 9 ⊢ (∪ (𝐻 “ ω) = (ℵ‘𝑧) → ¬ 𝑧 ∈ (ℵ‘𝑧))
41	7, 40	anim12i 614	. . . . . . . 8 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧)))
42		eloni 6322	. . . . . . . . . 10 ⊢ (𝑧 ∈ On → Ord 𝑧)
43	8	onordi 6425	. . . . . . . . . 10 ⊢ Ord (ℵ‘𝑧)
44		ordtri4 6349	. . . . . . . . . 10 ⊢ ((Ord 𝑧 ∧ Ord (ℵ‘𝑧)) → (𝑧 = (ℵ‘𝑧) ↔ (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧))))
45	42, 43, 44	sylancl 587	. . . . . . . . 9 ⊢ (𝑧 ∈ On → (𝑧 = (ℵ‘𝑧) ↔ (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧))))
46	45	adantr 480	. . . . . . . 8 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → (𝑧 = (ℵ‘𝑧) ↔ (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧))))
47	41, 46	mpbird 257	. . . . . . 7 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → 𝑧 = (ℵ‘𝑧))
48		eqeq2 2747	. . . . . . . 8 ⊢ (∪ (𝐻 “ ω) = (ℵ‘𝑧) → (𝑧 = ∪ (𝐻 “ ω) ↔ 𝑧 = (ℵ‘𝑧)))
49	48	adantl 481	. . . . . . 7 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → (𝑧 = ∪ (𝐻 “ ω) ↔ 𝑧 = (ℵ‘𝑧)))
50	47, 49	mpbird 257	. . . . . 6 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → 𝑧 = ∪ (𝐻 “ ω))
51	50	eqcomd 2741	. . . . 5 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → ∪ (𝐻 “ ω) = 𝑧)
52	51	fveq2d 6833	. . . 4 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → (ℵ‘∪ (𝐻 “ ω)) = (ℵ‘𝑧))
53		eqeq2 2747	. . . . 5 ⊢ (∪ (𝐻 “ ω) = (ℵ‘𝑧) → ((ℵ‘∪ (𝐻 “ ω)) = ∪ (𝐻 “ ω) ↔ (ℵ‘∪ (𝐻 “ ω)) = (ℵ‘𝑧)))
54	53	adantl 481	. . . 4 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → ((ℵ‘∪ (𝐻 “ ω)) = ∪ (𝐻 “ ω) ↔ (ℵ‘∪ (𝐻 “ ω)) = (ℵ‘𝑧)))
55	52, 54	mpbird 257	. . 3 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → (ℵ‘∪ (𝐻 “ ω)) = ∪ (𝐻 “ ω))
56	55	rexlimiva 3128	. 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 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3059   ⊆ wss 3885  ∪ cuni 4840  ran crn 5621   ↾ cres 5622   “ cima 5623  Ord word 6311  Oncon0 6312  suc csuc 6314  Fun wfun 6481   Fn wfn 6482  ‘cfv 6487  ωcom 7806  reccrdg 8337  cardccrd 9848  ℵcale 9849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678  ax-inf2 9551

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-rmo 3340  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-se 5574  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-isom 6496  df-riota 7313  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-er 8632  df-en 8883  df-dom 8884  df-sdom 8885  df-fin 8886  df-oi 9414  df-har 9461  df-card 9852  df-aleph 9853

This theorem is referenced by:  alephfp2  10020