Theorem domalom 37368

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 3266	. . . 4 ⊢ Ⅎ𝑛∀𝑛 ∈ ω 𝑛 ≼ 𝐴
2		breq1 5122	. . . . . . 7 ⊢ (𝑦 = 𝑛 → (𝑦 ≺ 𝐴 ↔ 𝑛 ≺ 𝐴))
3	2	imbi2d 340	. . . . . 6 ⊢ (𝑦 = 𝑛 → ((∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → 𝑦 ≺ 𝐴) ↔ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → 𝑛 ≺ 𝐴)))
4		breq1 5122	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝑦 ≺ 𝐴 ↔ ∅ ≺ 𝐴))
5		breq1 5122	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ≺ 𝐴 ↔ 𝑧 ≺ 𝐴))
6		breq1 5122	. . . . . . 7 ⊢ (𝑦 = suc 𝑧 → (𝑦 ≺ 𝐴 ↔ suc 𝑧 ≺ 𝐴))
7		1n0 8498	. . . . . . . . 9 ⊢ 1o ≠ ∅
8		1onn 8650	. . . . . . . . . 10 ⊢ 1o ∈ ω
9		0sdomg 9116	. . . . . . . . . 10 ⊢ (1o ∈ ω → (∅ ≺ 1o ↔ 1o ≠ ∅))
10	8, 9	ax-mp 5	. . . . . . . . 9 ⊢ (∅ ≺ 1o ↔ 1o ≠ ∅)
11	7, 10	mpbir 231	. . . . . . . 8 ⊢ ∅ ≺ 1o
12		breq1 5122	. . . . . . . . . 10 ⊢ (𝑛 = 1o → (𝑛 ≼ 𝐴 ↔ 1o ≼ 𝐴))
13	12	rspccv 3598	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → (1o ∈ ω → 1o ≼ 𝐴))
14	8, 13	mpi 20	. . . . . . . 8 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → 1o ≼ 𝐴)
15		sdomdomtr 9122	. . . . . . . 8 ⊢ ((∅ ≺ 1o ∧ 1o ≼ 𝐴) → ∅ ≺ 𝐴)
16	11, 14, 15	sylancr 587	. . . . . . 7 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → ∅ ≺ 𝐴)
17		peano2 7884	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ω → suc 𝑧 ∈ ω)
18		php4 9222	. . . . . . . . . . 11 ⊢ (suc 𝑧 ∈ ω → suc 𝑧 ≺ suc suc 𝑧)
19	17, 18	syl 17	. . . . . . . . . 10 ⊢ (𝑧 ∈ ω → suc 𝑧 ≺ suc suc 𝑧)
20		breq1 5122	. . . . . . . . . . . 12 ⊢ (𝑛 = suc suc 𝑧 → (𝑛 ≼ 𝐴 ↔ suc suc 𝑧 ≼ 𝐴))
21	20	rspccv 3598	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → (suc suc 𝑧 ∈ ω → suc suc 𝑧 ≼ 𝐴))
22		peano2 7884	. . . . . . . . . . . 12 ⊢ (suc 𝑧 ∈ ω → suc suc 𝑧 ∈ ω)
23	17, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ω → suc suc 𝑧 ∈ ω)
24	21, 23	impel 505	. . . . . . . . . 10 ⊢ ((∀𝑛 ∈ ω 𝑛 ≼ 𝐴 ∧ 𝑧 ∈ ω) → suc suc 𝑧 ≼ 𝐴)
25		sdomdomtr 9122	. . . . . . . . . 10 ⊢ ((suc 𝑧 ≺ suc suc 𝑧 ∧ suc suc 𝑧 ≼ 𝐴) → suc 𝑧 ≺ 𝐴)
26	19, 24, 25	syl2an2 686	. . . . . . . . 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 413	. . . . . . 7 ⊢ (𝑧 ∈ ω → (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → (𝑧 ≺ 𝐴 → suc 𝑧 ≺ 𝐴)))
29	4, 5, 6, 16, 28	finds2 7892	. . . . . 6 ⊢ (𝑦 ∈ ω → (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → 𝑦 ≺ 𝐴))
30	3, 29	vtoclga 3556	. . . . 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 3240	. . 3 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → ∀𝑛 ∈ ω 𝑛 ≺ 𝐴)
33		sdomnen 8993	. . . . 5 ⊢ (𝑛 ≺ 𝐴 → ¬ 𝑛 ≈ 𝐴)
34		ensym 9015	. . . . 5 ⊢ (𝐴 ≈ 𝑛 → 𝑛 ≈ 𝐴)
35	33, 34	nsyl 140	. . . 4 ⊢ (𝑛 ≺ 𝐴 → ¬ 𝐴 ≈ 𝑛)
36	35	ralimi 3073	. . 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 8988	. . . 4 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ω 𝐴 ≈ 𝑛)
39	38	notbii 320	. . 3 ⊢ (¬ 𝐴 ∈ Fin ↔ ¬ ∃𝑛 ∈ ω 𝐴 ≈ 𝑛)
40		ralnex 3062	. . 3 ⊢ (∀𝑛 ∈ ω ¬ 𝐴 ≈ 𝑛 ↔ ¬ ∃𝑛 ∈ ω 𝐴 ≈ 𝑛)
41	39, 40	bitr4i 278	. 2 ⊢ (¬ 𝐴 ∈ Fin ↔ ∀𝑛 ∈ ω ¬ 𝐴 ≈ 𝑛)
42	37, 41	sylibr 234	1 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → ¬ 𝐴 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  ∅c0 4308   class class class wbr 5119  suc csuc 6354  ωcom 7859  1_oc1o 8471   ≈ cen 8954   ≼ cdom 8955   ≺ csdm 8956  Fincfn 8957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-om 7860  df-1o 8478  df-er 8717  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961

This theorem is referenced by:  isinf2  37369