Theorem sdom2en01 10215

Description: A set with less than two elements has 0 or 1. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Assertion

Ref	Expression
sdom2en01	⊢ (𝐴 ≺ 2o ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1o))

Proof of Theorem sdom2en01

Step	Hyp	Ref	Expression
1		onfin2 9141	. . . . 5 ⊢ ω = (On ∩ Fin)
2		inss2 4166	. . . . 5 ⊢ (On ∩ Fin) ⊆ Fin
3	1, 2	eqsstri 3961	. . . 4 ⊢ ω ⊆ Fin
4		2onn 8568	. . . 4 ⊢ 2o ∈ ω
5	3, 4	sselii 3912	. . 3 ⊢ 2o ∈ Fin
6		sdomdom 8917	. . 3 ⊢ (𝐴 ≺ 2o → 𝐴 ≼ 2o)
7		domfi 9113	. . 3 ⊢ ((2o ∈ Fin ∧ 𝐴 ≼ 2o) → 𝐴 ∈ Fin)
8	5, 6, 7	sylancr 593	. 2 ⊢ (𝐴 ≺ 2o → 𝐴 ∈ Fin)
9		id 22	. . . 4 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
10		0fi 8979	. . . 4 ⊢ ∅ ∈ Fin
11	9, 10	eqeltrdi 2847	. . 3 ⊢ (𝐴 = ∅ → 𝐴 ∈ Fin)
12		1onn 8566	. . . . 5 ⊢ 1o ∈ ω
13	3, 12	sselii 3912	. . . 4 ⊢ 1o ∈ Fin
14		enfi 9111	. . . 4 ⊢ (𝐴 ≈ 1o → (𝐴 ∈ Fin ↔ 1o ∈ Fin))
15	13, 14	mpbiri 259	. . 3 ⊢ (𝐴 ≈ 1o → 𝐴 ∈ Fin)
16	11, 15	jaoi 863	. 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 ≈ 1o) → 𝐴 ∈ Fin)
17		df2o3 8403	. . . . . 6 ⊢ 2o = {∅, 1o}
18	17	eleq2i 2831	. . . . 5 ⊢ ((card‘𝐴) ∈ 2o ↔ (card‘𝐴) ∈ {∅, 1o})
19		fvex 6840	. . . . . 6 ⊢ (card‘𝐴) ∈ V
20	19	elpr 4580	. . . . 5 ⊢ ((card‘𝐴) ∈ {∅, 1o} ↔ ((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1o))
21	18, 20	bitri 276	. . . 4 ⊢ ((card‘𝐴) ∈ 2o ↔ ((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1o))
22	21	a1i 11	. . 3 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) ∈ 2o ↔ ((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1o)))
23		cardnn 9878	. . . . . 6 ⊢ (2o ∈ ω → (card‘2o) = 2o)
24	4, 23	ax-mp 5	. . . . 5 ⊢ (card‘2o) = 2o
25	24	eleq2i 2831	. . . 4 ⊢ ((card‘𝐴) ∈ (card‘2o) ↔ (card‘𝐴) ∈ 2o)
26		finnum 9863	. . . . 5 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card)
27		2on 8408	. . . . . 6 ⊢ 2o ∈ On
28		onenon 9864	. . . . . 6 ⊢ (2o ∈ On → 2o ∈ dom card)
29	27, 28	ax-mp 5	. . . . 5 ⊢ 2o ∈ dom card
30		cardsdom2 9903	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 2o ∈ dom card) → ((card‘𝐴) ∈ (card‘2o) ↔ 𝐴 ≺ 2o))
31	26, 29, 30	sylancl 592	. . . 4 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) ∈ (card‘2o) ↔ 𝐴 ≺ 2o))
32	25, 31	bitr3id 286	. . 3 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) ∈ 2o ↔ 𝐴 ≺ 2o))
33		cardnueq0 9879	. . . . 5 ⊢ (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
34	26, 33	syl 17	. . . 4 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
35		cardnn 9878	. . . . . . 7 ⊢ (1o ∈ ω → (card‘1o) = 1o)
36	12, 35	ax-mp 5	. . . . . 6 ⊢ (card‘1o) = 1o
37	36	eqeq2i 2752	. . . . 5 ⊢ ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o)
38		finnum 9863	. . . . . . 7 ⊢ (1o ∈ Fin → 1o ∈ dom card)
39	13, 38	ax-mp 5	. . . . . 6 ⊢ 1o ∈ dom card
40		carden2 9902	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ 1o ∈ dom card) → ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o))
41	26, 39, 40	sylancl 592	. . . . 5 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o))
42	37, 41	bitr3id 286	. . . 4 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) = 1o ↔ 𝐴 ≈ 1o))
43	34, 42	orbi12d 924	. . 3 ⊢ (𝐴 ∈ Fin → (((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1o) ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1o)))
44	22, 32, 43	3bitr3d 310	. 2 ⊢ (𝐴 ∈ Fin → (𝐴 ≺ 2o ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1o)))
45	8, 16, 44	pm5.21nii 379	1 ⊢ (𝐴 ≺ 2o ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1o))

Syntax hints:   ↔ wb 207   ∨ wo 853   = wceq 1547   ∈ wcel 2119   ∩ cin 3882  ∅c0 4261  {cpr 4557   class class class wbr 5072  dom cdm 5618  Oncon0 6310  ‘cfv 6485  ωcom 7806  1_oc1o 8388  2_oc2o 8389   ≈ cen 8880   ≼ cdom 8881   ≺ csdm 8882  Fincfn 8883  cardccrd 9850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-om 7807  df-1o 8395  df-2o 8396  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9854

This theorem is referenced by:  fin56  10306  en2top  22968