Theorem domalom 35502

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 3142	. . . 4 ⊢ Ⅎ𝑛∀𝑛 ∈ ω 𝑛 ≼ 𝐴
2		breq1 5073	. . . . . . 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 5073	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝑦 ≺ 𝐴 ↔ ∅ ≺ 𝐴))
5		breq1 5073	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ≺ 𝐴 ↔ 𝑧 ≺ 𝐴))
6		breq1 5073	. . . . . . 7 ⊢ (𝑦 = suc 𝑧 → (𝑦 ≺ 𝐴 ↔ suc 𝑧 ≺ 𝐴))
7		1n0 8286	. . . . . . . . 9 ⊢ 1o ≠ ∅
8		1onn 8432	. . . . . . . . . 10 ⊢ 1o ∈ ω
9		0sdomg 8842	. . . . . . . . . 10 ⊢ (1o ∈ ω → (∅ ≺ 1o ↔ 1o ≠ ∅))
10	8, 9	ax-mp 5	. . . . . . . . 9 ⊢ (∅ ≺ 1o ↔ 1o ≠ ∅)
11	7, 10	mpbir 230	. . . . . . . 8 ⊢ ∅ ≺ 1o
12		breq1 5073	. . . . . . . . . 10 ⊢ (𝑛 = 1o → (𝑛 ≼ 𝐴 ↔ 1o ≼ 𝐴))
13	12	rspccv 3549	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → (1o ∈ ω → 1o ≼ 𝐴))
14	8, 13	mpi 20	. . . . . . . 8 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → 1o ≼ 𝐴)
15		sdomdomtr 8846	. . . . . . . 8 ⊢ ((∅ ≺ 1o ∧ 1o ≼ 𝐴) → ∅ ≺ 𝐴)
16	11, 14, 15	sylancr 586	. . . . . . 7 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → ∅ ≺ 𝐴)
17		peano2 7711	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ω → suc 𝑧 ∈ ω)
18		php4 8900	. . . . . . . . . . 11 ⊢ (suc 𝑧 ∈ ω → suc 𝑧 ≺ suc suc 𝑧)
19	17, 18	syl 17	. . . . . . . . . 10 ⊢ (𝑧 ∈ ω → suc 𝑧 ≺ suc suc 𝑧)
20		breq1 5073	. . . . . . . . . . . 12 ⊢ (𝑛 = suc suc 𝑧 → (𝑛 ≼ 𝐴 ↔ suc suc 𝑧 ≼ 𝐴))
21	20	rspccv 3549	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → (suc suc 𝑧 ∈ ω → suc suc 𝑧 ≼ 𝐴))
22		peano2 7711	. . . . . . . . . . . 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 8846	. . . . . . . . . 10 ⊢ ((suc 𝑧 ≺ suc suc 𝑧 ∧ suc suc 𝑧 ≼ 𝐴) → suc 𝑧 ≺ 𝐴)
26	19, 24, 25	syl2an2 682	. . . . . . . . 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 7721	. . . . . 6 ⊢ (𝑦 ∈ ω → (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → 𝑦 ≺ 𝐴))
30	3, 29	vtoclga 3503	. . . . 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 3139	. . 3 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → ∀𝑛 ∈ ω 𝑛 ≺ 𝐴)
33		sdomnen 8724	. . . . 5 ⊢ (𝑛 ≺ 𝐴 → ¬ 𝑛 ≈ 𝐴)
34		ensym 8744	. . . . 5 ⊢ (𝐴 ≈ 𝑛 → 𝑛 ≈ 𝐴)
35	33, 34	nsyl 140	. . . 4 ⊢ (𝑛 ≺ 𝐴 → ¬ 𝐴 ≈ 𝑛)
36	35	ralimi 3086	. . 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 8719	. . . 4 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ω 𝐴 ≈ 𝑛)
39	38	notbii 319	. . 3 ⊢ (¬ 𝐴 ∈ Fin ↔ ¬ ∃𝑛 ∈ ω 𝐴 ≈ 𝑛)
40		ralnex 3163	. . 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 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  ∅c0 4253   class class class wbr 5070  suc csuc 6253  ωcom 7687  1_oc1o 8260   ≈ cen 8688   ≼ cdom 8689   ≺ csdm 8690  Fincfn 8691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-om 7688  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695

This theorem is referenced by:  isinf2  35503