Theorem domalom 35575

Description: A class which dominates every natural number is not finite. (Contributed by ML, 14-Dec-2020.)

Assertion

Ref	Expression
domalom	⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → ¬ 𝐴 ∈ Fin)

Distinct variable group:   𝐴,𝑛

Proof of Theorem domalom

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

Step	Hyp	Ref	Expression
1		nfra1 3144	. . . 4 ⊢ Ⅎ𝑛∀𝑛 ∈ ω 𝑛 ≼ 𝐴
2		breq1 5077	. . . . . . 7 ⊢ (𝑦 = 𝑛 → (𝑦 ≺ 𝐴 ↔ 𝑛 ≺ 𝐴))
3	2	imbi2d 341	. . . . . 6 ⊢ (𝑦 = 𝑛 → ((∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → 𝑦 ≺ 𝐴) ↔ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → 𝑛 ≺ 𝐴)))
4		breq1 5077	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝑦 ≺ 𝐴 ↔ ∅ ≺ 𝐴))
5		breq1 5077	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ≺ 𝐴 ↔ 𝑧 ≺ 𝐴))
6		breq1 5077	. . . . . . 7 ⊢ (𝑦 = suc 𝑧 → (𝑦 ≺ 𝐴 ↔ suc 𝑧 ≺ 𝐴))
7		1n0 8318	. . . . . . . . 9 ⊢ 1o ≠ ∅
8		1onn 8470	. . . . . . . . . 10 ⊢ 1o ∈ ω
9		0sdomg 8891	. . . . . . . . . 10 ⊢ (1o ∈ ω → (∅ ≺ 1o ↔ 1o ≠ ∅))
10	8, 9	ax-mp 5	. . . . . . . . 9 ⊢ (∅ ≺ 1o ↔ 1o ≠ ∅)
11	7, 10	mpbir 230	. . . . . . . 8 ⊢ ∅ ≺ 1o
12		breq1 5077	. . . . . . . . . 10 ⊢ (𝑛 = 1o → (𝑛 ≼ 𝐴 ↔ 1o ≼ 𝐴))
13	12	rspccv 3558	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → (1o ∈ ω → 1o ≼ 𝐴))
14	8, 13	mpi 20	. . . . . . . 8 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → 1o ≼ 𝐴)
15		sdomdomtr 8897	. . . . . . . 8 ⊢ ((∅ ≺ 1o ∧ 1o ≼ 𝐴) → ∅ ≺ 𝐴)
16	11, 14, 15	sylancr 587	. . . . . . 7 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → ∅ ≺ 𝐴)
17		peano2 7737	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ω → suc 𝑧 ∈ ω)
18		php4 8996	. . . . . . . . . . 11 ⊢ (suc 𝑧 ∈ ω → suc 𝑧 ≺ suc suc 𝑧)
19	17, 18	syl 17	. . . . . . . . . 10 ⊢ (𝑧 ∈ ω → suc 𝑧 ≺ suc suc 𝑧)
20		breq1 5077	. . . . . . . . . . . 12 ⊢ (𝑛 = suc suc 𝑧 → (𝑛 ≼ 𝐴 ↔ suc suc 𝑧 ≼ 𝐴))
21	20	rspccv 3558	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → (suc suc 𝑧 ∈ ω → suc suc 𝑧 ≼ 𝐴))
22		peano2 7737	. . . . . . . . . . . 12 ⊢ (suc 𝑧 ∈ ω → suc suc 𝑧 ∈ ω)
23	17, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ω → suc suc 𝑧 ∈ ω)
24	21, 23	impel 506	. . . . . . . . . 10 ⊢ ((∀𝑛 ∈ ω 𝑛 ≼ 𝐴 ∧ 𝑧 ∈ ω) → suc suc 𝑧 ≼ 𝐴)
25		sdomdomtr 8897	. . . . . . . . . 10 ⊢ ((suc 𝑧 ≺ suc suc 𝑧 ∧ suc suc 𝑧 ≼ 𝐴) → suc 𝑧 ≺ 𝐴)
26	19, 24, 25	syl2an2 683	. . . . . . . . 9 ⊢ ((∀𝑛 ∈ ω 𝑛 ≼ 𝐴 ∧ 𝑧 ∈ ω) → suc 𝑧 ≺ 𝐴)
27	26	a1d 25	. . . . . . . 8 ⊢ ((∀𝑛 ∈ ω 𝑛 ≼ 𝐴 ∧ 𝑧 ∈ ω) → (𝑧 ≺ 𝐴 → suc 𝑧 ≺ 𝐴))
28	27	expcom 414	. . . . . . 7 ⊢ (𝑧 ∈ ω → (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → (𝑧 ≺ 𝐴 → suc 𝑧 ≺ 𝐴)))
29	4, 5, 6, 16, 28	finds2 7747	. . . . . 6 ⊢ (𝑦 ∈ ω → (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → 𝑦 ≺ 𝐴))
30	3, 29	vtoclga 3513	. . . . 5 ⊢ (𝑛 ∈ ω → (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → 𝑛 ≺ 𝐴))
31	30	com12 32	. . . 4 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → (𝑛 ∈ ω → 𝑛 ≺ 𝐴))
32	1, 31	ralrimi 3141	. . 3 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → ∀𝑛 ∈ ω 𝑛 ≺ 𝐴)
33		sdomnen 8769	. . . . 5 ⊢ (𝑛 ≺ 𝐴 → ¬ 𝑛 ≈ 𝐴)
34		ensym 8789	. . . . 5 ⊢ (𝐴 ≈ 𝑛 → 𝑛 ≈ 𝐴)
35	33, 34	nsyl 140	. . . 4 ⊢ (𝑛 ≺ 𝐴 → ¬ 𝐴 ≈ 𝑛)
36	35	ralimi 3087	. . 3 ⊢ (∀𝑛 ∈ ω 𝑛 ≺ 𝐴 → ∀𝑛 ∈ ω ¬ 𝐴 ≈ 𝑛)
37	32, 36	syl 17	. 2 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → ∀𝑛 ∈ ω ¬ 𝐴 ≈ 𝑛)
38		isfi 8764	. . . 4 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ω 𝐴 ≈ 𝑛)
39	38	notbii 320	. . 3 ⊢ (¬ 𝐴 ∈ Fin ↔ ¬ ∃𝑛 ∈ ω 𝐴 ≈ 𝑛)
40		ralnex 3167	. . 3 ⊢ (∀𝑛 ∈ ω ¬ 𝐴 ≈ 𝑛 ↔ ¬ ∃𝑛 ∈ ω 𝐴 ≈ 𝑛)
41	39, 40	bitr4i 277	. 2 ⊢ (¬ 𝐴 ∈ Fin ↔ ∀𝑛 ∈ ω ¬ 𝐴 ≈ 𝑛)
42	37, 41	sylibr 233	1 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → ¬ 𝐴 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  ∅c0 4256   class class class wbr 5074  suc csuc 6268  ωcom 7712  1_oc1o 8290   ≈ cen 8730   ≼ cdom 8731   ≺ csdm 8732  Fincfn 8733

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-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

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-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-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-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-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-om 7713  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737

This theorem is referenced by:  isinf2  35576