Theorem alephfp 10158

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 10159 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 10157	. 2 ⊢ ∪ (𝐻 “ ω) ∈ ran ℵ
3		isinfcard 10142	. . 3 ⊢ ((ω ⊆ ∪ (𝐻 “ ω) ∧ (card‘∪ (𝐻 “ ω)) = ∪ (𝐻 “ ω)) ↔ ∪ (𝐻 “ ω) ∈ ran ℵ)
4		cardalephex 10140	. . . 4 ⊢ (ω ⊆ ∪ (𝐻 “ ω) → ((card‘∪ (𝐻 “ ω)) = ∪ (𝐻 “ ω) ↔ ∃𝑧 ∈ On ∪ (𝐻 “ ω) = (ℵ‘𝑧)))
5	4	biimpa 482	. . 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 10138	. . . . . . . . 9 ⊢ (𝑧 ∈ On → 𝑧 ⊆ (ℵ‘𝑧))
8		alephon 10119	. . . . . . . . . . 11 ⊢ (ℵ‘𝑧) ∈ On
9	8	onirri 6466	. . . . . . . . . 10 ⊢ ¬ (ℵ‘𝑧) ∈ (ℵ‘𝑧)
10		frfnom 8421	. . . . . . . . . . . . . 14 ⊢ (rec(ℵ, ω) ↾ ω) Fn ω
11	1	fneq1i 6624	. . . . . . . . . . . . . 14 ⊢ (𝐻 Fn ω ↔ (rec(ℵ, ω) ↾ ω) Fn ω)
12	10, 11	mpbir 234	. . . . . . . . . . . . 13 ⊢ 𝐻 Fn ω
13		fnfun 6627	. . . . . . . . . . . . 13 ⊢ (𝐻 Fn ω → Fun 𝐻)
14		eluniima 7242	. . . . . . . . . . . . 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 10150	. . . . . . . . . . . . . . . 16 ⊢ ran ℵ ⊆ On
17	1	alephfplem3 10156	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ ω → (𝐻‘𝑣) ∈ ran ℵ)
18	16, 17	sselid 3928	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ ω → (𝐻‘𝑣) ∈ On)
19		alephord2i 10127	. . . . . . . . . . . . . . 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 10155	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ ω → (𝐻‘suc 𝑣) = (ℵ‘(𝐻‘𝑣)))
22		peano2 7884	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ∈ ω → suc 𝑣 ∈ ω)
23		fnfvelrn 7068	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐻 Fn ω ∧ suc 𝑣 ∈ ω) → (𝐻‘suc 𝑣) ∈ ran 𝐻)
24	12, 23	mpan 703	. . . . . . . . . . . . . . . . . . 19 ⊢ (suc 𝑣 ∈ ω → (𝐻‘suc 𝑣) ∈ ran 𝐻)
25		fnima 6657	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐻 Fn ω → (𝐻 “ ω) = ran 𝐻)
26	12, 25	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐻 “ ω) = ran 𝐻
27	24, 26	eleqtrrdi 2871	. . . . . . . . . . . . . . . . . 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 2861	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ ω → (ℵ‘(𝐻‘𝑣)) ∈ (𝐻 “ ω))
30		elssuni 4898	. . . . . . . . . . . . . . . 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 3929	. . . . . . . . . . . . . 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 3156	. . . . . . . . . . . 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 2849	. . . . . . . . . . . 12 ⊢ (∪ (𝐻 “ ω) = (ℵ‘𝑧) → (𝑧 ∈ ∪ (𝐻 “ ω) ↔ 𝑧 ∈ (ℵ‘𝑧)))
37		eleq2 2849	. . . . . . . . . . . 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 625	. . . . . . . 8 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧)))
42		eloni 6361	. . . . . . . . . 10 ⊢ (𝑧 ∈ On → Ord 𝑧)
43	8	onordi 6465	. . . . . . . . . 10 ⊢ Ord (ℵ‘𝑧)
44		ordtri4 6389	. . . . . . . . . 10 ⊢ ((Ord 𝑧 ∧ Ord (ℵ‘𝑧)) → (𝑧 = (ℵ‘𝑧) ↔ (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧))))
45	42, 43, 44	sylancl 598	. . . . . . . . 9 ⊢ (𝑧 ∈ On → (𝑧 = (ℵ‘𝑧) ↔ (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧))))
46	45	adantr 486	. . . . . . . 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 2772	. . . . . . . 8 ⊢ (∪ (𝐻 “ ω) = (ℵ‘𝑧) → (𝑧 = ∪ (𝐻 “ ω) ↔ 𝑧 = (ℵ‘𝑧)))
49	48	adantl 487	. . . . . . 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 2766	. . . . 5 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → ∪ (𝐻 “ ω) = 𝑧)
52	51	fveq2d 6877	. . . 4 ⊢ ((𝑧 ∈ On ∧ ∪ (𝐻 “ ω) = (ℵ‘𝑧)) → (ℵ‘∪ (𝐻 “ ω)) = (ℵ‘𝑧))
53		eqeq2 2772	. . . . 5 ⊢ (∪ (𝐻 “ ω) = (ℵ‘𝑧) → ((ℵ‘∪ (𝐻 “ ω)) = ∪ (𝐻 “ ω) ↔ (ℵ‘∪ (𝐻 “ ω)) = (ℵ‘𝑧)))
54	53	adantl 487	. . . 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 3155	. 2 ⊢ (∃𝑧 ∈ On ∪ (𝐻 “ ω) = (ℵ‘𝑧) → (ℵ‘∪ (𝐻 “ ω)) = ∪ (𝐻 “ ω))
57	2, 6, 56	mp2b 10	1 ⊢ (ℵ‘∪ (𝐻 “ ω)) = ∪ (𝐻 “ ω)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3086   ⊆ wss 3898  ∪ cuni 4866  ran crn 5648   ↾ cres 5649   “ cima 5650  Ord word 6350  Oncon0 6351  suc csuc 6353  Fun wfun 6521   Fn wfn 6522  ‘cfv 6527  ωcom 7860  reccrdg 8395  cardccrd 9987  ℵcale 9988

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-inf2 9620

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-isom 6536  df-riota 7365  df-ov 7411  df-om 7861  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-er 8695  df-en 8952  df-dom 8953  df-sdom 8954  df-fin 8955  df-oi 9482  df-har 9529  df-card 9991  df-aleph 9992

This theorem is used by:  alephfp2  10159