Theorem domalom 37392

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 3261	. . . 4 ⊢ Ⅎ𝑛∀𝑛 ∈ ω 𝑛 ≼ 𝐴
2		breq1 5110	. . . . . . 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 5110	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝑦 ≺ 𝐴 ↔ ∅ ≺ 𝐴))
5		breq1 5110	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ≺ 𝐴 ↔ 𝑧 ≺ 𝐴))
6		breq1 5110	. . . . . . 7 ⊢ (𝑦 = suc 𝑧 → (𝑦 ≺ 𝐴 ↔ suc 𝑧 ≺ 𝐴))
7		1n0 8452	. . . . . . . . 9 ⊢ 1o ≠ ∅
8		1onn 8604	. . . . . . . . . 10 ⊢ 1o ∈ ω
9		0sdomg 9070	. . . . . . . . . 10 ⊢ (1o ∈ ω → (∅ ≺ 1o ↔ 1o ≠ ∅))
10	8, 9	ax-mp 5	. . . . . . . . 9 ⊢ (∅ ≺ 1o ↔ 1o ≠ ∅)
11	7, 10	mpbir 231	. . . . . . . 8 ⊢ ∅ ≺ 1o
12		breq1 5110	. . . . . . . . . 10 ⊢ (𝑛 = 1o → (𝑛 ≼ 𝐴 ↔ 1o ≼ 𝐴))
13	12	rspccv 3585	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → (1o ∈ ω → 1o ≼ 𝐴))
14	8, 13	mpi 20	. . . . . . . 8 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → 1o ≼ 𝐴)
15		sdomdomtr 9074	. . . . . . . 8 ⊢ ((∅ ≺ 1o ∧ 1o ≼ 𝐴) → ∅ ≺ 𝐴)
16	11, 14, 15	sylancr 587	. . . . . . 7 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → ∅ ≺ 𝐴)
17		peano2 7866	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ω → suc 𝑧 ∈ ω)
18		php4 9174	. . . . . . . . . . 11 ⊢ (suc 𝑧 ∈ ω → suc 𝑧 ≺ suc suc 𝑧)
19	17, 18	syl 17	. . . . . . . . . 10 ⊢ (𝑧 ∈ ω → suc 𝑧 ≺ suc suc 𝑧)
20		breq1 5110	. . . . . . . . . . . 12 ⊢ (𝑛 = suc suc 𝑧 → (𝑛 ≼ 𝐴 ↔ suc suc 𝑧 ≼ 𝐴))
21	20	rspccv 3585	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → (suc suc 𝑧 ∈ ω → suc suc 𝑧 ≼ 𝐴))
22		peano2 7866	. . . . . . . . . . . 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 9074	. . . . . . . . . 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 7874	. . . . . 6 ⊢ (𝑦 ∈ ω → (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → 𝑦 ≺ 𝐴))
30	3, 29	vtoclga 3543	. . . . 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 3235	. . 3 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → ∀𝑛 ∈ ω 𝑛 ≺ 𝐴)
33		sdomnen 8952	. . . . 5 ⊢ (𝑛 ≺ 𝐴 → ¬ 𝑛 ≈ 𝐴)
34		ensym 8974	. . . . 5 ⊢ (𝐴 ≈ 𝑛 → 𝑛 ≈ 𝐴)
35	33, 34	nsyl 140	. . . 4 ⊢ (𝑛 ≺ 𝐴 → ¬ 𝐴 ≈ 𝑛)
36	35	ralimi 3066	. . 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 8947	. . . 4 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ω 𝐴 ≈ 𝑛)
39	38	notbii 320	. . 3 ⊢ (¬ 𝐴 ∈ Fin ↔ ¬ ∃𝑛 ∈ ω 𝐴 ≈ 𝑛)
40		ralnex 3055	. . 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 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  ∅c0 4296   class class class wbr 5107  suc csuc 6334  ωcom 7842  1_oc1o 8427   ≈ cen 8915   ≼ cdom 8916   ≺ csdm 8917  Fincfn 8918

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-om 7843  df-1o 8434  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922

This theorem is referenced by:  isinf2  37393