Theorem domalom 37899

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 3287	. . . 4 ⊢ Ⅎ𝑛∀𝑛 ∈ ω 𝑛 ≼ 𝐴
2		breq1 5104	. . . . . . 7 ⊢ (𝑦 = 𝑛 → (𝑦 ≺ 𝐴 ↔ 𝑛 ≺ 𝐴))
3	2	imbi2d 342	. . . . . 6 ⊢ (𝑦 = 𝑛 → ((∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → 𝑦 ≺ 𝐴) ↔ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → 𝑛 ≺ 𝐴)))
4		breq1 5104	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝑦 ≺ 𝐴 ↔ ∅ ≺ 𝐴))
5		breq1 5104	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ≺ 𝐴 ↔ 𝑧 ≺ 𝐴))
6		breq1 5104	. . . . . . 7 ⊢ (𝑦 = suc 𝑧 → (𝑦 ≺ 𝐴 ↔ suc 𝑧 ≺ 𝐴))
7		1n0 8457	. . . . . . . . 9 ⊢ 1o ≠ ∅
8		1onn 8611	. . . . . . . . . 10 ⊢ 1o ∈ ω
9		0sdomg 9079	. . . . . . . . . 10 ⊢ (1o ∈ ω → (∅ ≺ 1o ↔ 1o ≠ ∅))
10	8, 9	ax-mp 5	. . . . . . . . 9 ⊢ (∅ ≺ 1o ↔ 1o ≠ ∅)
11	7, 10	mpbir 233	. . . . . . . 8 ⊢ ∅ ≺ 1o
12		breq1 5104	. . . . . . . . . 10 ⊢ (𝑛 = 1o → (𝑛 ≼ 𝐴 ↔ 1o ≼ 𝐴))
13	12	rspccv 3579	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → (1o ∈ ω → 1o ≼ 𝐴))
14	8, 13	mpi 20	. . . . . . . 8 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → 1o ≼ 𝐴)
15		sdomdomtr 9083	. . . . . . . 8 ⊢ ((∅ ≺ 1o ∧ 1o ≼ 𝐴) → ∅ ≺ 𝐴)
16	11, 14, 15	sylancr 596	. . . . . . 7 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → ∅ ≺ 𝐴)
17		peano2 7871	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ω → suc 𝑧 ∈ ω)
18		php4 9179	. . . . . . . . . . 11 ⊢ (suc 𝑧 ∈ ω → suc 𝑧 ≺ suc suc 𝑧)
19	17, 18	syl 17	. . . . . . . . . 10 ⊢ (𝑧 ∈ ω → suc 𝑧 ≺ suc suc 𝑧)
20		breq1 5104	. . . . . . . . . . . 12 ⊢ (𝑛 = suc suc 𝑧 → (𝑛 ≼ 𝐴 ↔ suc suc 𝑧 ≼ 𝐴))
21	20	rspccv 3579	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → (suc suc 𝑧 ∈ ω → suc suc 𝑧 ≼ 𝐴))
22		peano2 7871	. . . . . . . . . . . 12 ⊢ (suc 𝑧 ∈ ω → suc suc 𝑧 ∈ ω)
23	17, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ω → suc suc 𝑧 ∈ ω)
24	21, 23	impel 513	. . . . . . . . . 10 ⊢ ((∀𝑛 ∈ ω 𝑛 ≼ 𝐴 ∧ 𝑧 ∈ ω) → suc suc 𝑧 ≼ 𝐴)
25		sdomdomtr 9083	. . . . . . . . . 10 ⊢ ((suc 𝑧 ≺ suc suc 𝑧 ∧ suc suc 𝑧 ≼ 𝐴) → suc 𝑧 ≺ 𝐴)
26	19, 24, 25	syl2an2 696	. . . . . . . . 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 417	. . . . . . 7 ⊢ (𝑧 ∈ ω → (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → (𝑧 ≺ 𝐴 → suc 𝑧 ≺ 𝐴)))
29	4, 5, 6, 16, 28	finds2 7880	. . . . . 6 ⊢ (𝑦 ∈ ω → (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → 𝑦 ≺ 𝐴))
30	3, 29	vtoclga 3542	. . . . 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 3261	. . 3 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → ∀𝑛 ∈ ω 𝑛 ≺ 𝐴)
33		sdomnen 8963	. . . . 5 ⊢ (𝑛 ≺ 𝐴 → ¬ 𝑛 ≈ 𝐴)
34		ensym 8985	. . . . 5 ⊢ (𝐴 ≈ 𝑛 → 𝑛 ≈ 𝐴)
35	33, 34	nsyl 140	. . . 4 ⊢ (𝑛 ≺ 𝐴 → ¬ 𝐴 ≈ 𝑛)
36	35	ralimi 3100	. . 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 8957	. . . 4 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ω 𝐴 ≈ 𝑛)
39	38	notbii 322	. . 3 ⊢ (¬ 𝐴 ∈ Fin ↔ ¬ ∃𝑛 ∈ ω 𝐴 ≈ 𝑛)
40		ralnex 3089	. . 3 ⊢ (∀𝑛 ∈ ω ¬ 𝐴 ≈ 𝑛 ↔ ¬ ∃𝑛 ∈ ω 𝐴 ≈ 𝑛)
41	39, 40	bitr4i 280	. 2 ⊢ (¬ 𝐴 ∈ Fin ↔ ∀𝑛 ∈ ω ¬ 𝐴 ≈ 𝑛)
42	37, 41	sylibr 236	1 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → ¬ 𝐴 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1561   ∈ wcel 2143   ≠ wne 2958  ∀wral 3077  ∃wrex 3087  ∅c0 4286   class class class wbr 5101  suc csuc 6349  ωcom 7847  1_oc1o 8431   ≈ cen 8925   ≼ cdom 8926   ≺ csdm 8927  Fincfn 8928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-om 7848  df-1o 8438  df-er 8679  df-en 8929  df-dom 8930  df-sdom 8931  df-fin 8932

This theorem is referenced by:  isinf2  37900