Theorem sdom2en01 9776

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 8763	. . . . 5 ⊢ ω = (On ∩ Fin)
2		inss2 4137	. . . . 5 ⊢ (On ∩ Fin) ⊆ Fin
3	1, 2	eqsstri 3929	. . . 4 ⊢ ω ⊆ Fin
4		2onn 8283	. . . 4 ⊢ 2o ∈ ω
5	3, 4	sselii 3892	. . 3 ⊢ 2o ∈ Fin
6		sdomdom 8569	. . 3 ⊢ (𝐴 ≺ 2o → 𝐴 ≼ 2o)
7		domfi 8790	. . 3 ⊢ ((2o ∈ Fin ∧ 𝐴 ≼ 2o) → 𝐴 ∈ Fin)
8	5, 6, 7	sylancr 590	. 2 ⊢ (𝐴 ≺ 2o → 𝐴 ∈ Fin)
9		id 22	. . . 4 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
10		0fin 8754	. . . 4 ⊢ ∅ ∈ Fin
11	9, 10	eqeltrdi 2861	. . 3 ⊢ (𝐴 = ∅ → 𝐴 ∈ Fin)
12		1onn 8282	. . . . 5 ⊢ 1o ∈ ω
13	3, 12	sselii 3892	. . . 4 ⊢ 1o ∈ Fin
14		enfi 8786	. . . 4 ⊢ (𝐴 ≈ 1o → (𝐴 ∈ Fin ↔ 1o ∈ Fin))
15	13, 14	mpbiri 261	. . 3 ⊢ (𝐴 ≈ 1o → 𝐴 ∈ Fin)
16	11, 15	jaoi 854	. 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 ≈ 1o) → 𝐴 ∈ Fin)
17		df2o3 8134	. . . . . 6 ⊢ 2o = {∅, 1o}
18	17	eleq2i 2844	. . . . 5 ⊢ ((card‘𝐴) ∈ 2o ↔ (card‘𝐴) ∈ {∅, 1o})
19		fvex 6677	. . . . . 6 ⊢ (card‘𝐴) ∈ V
20	19	elpr 4549	. . . . 5 ⊢ ((card‘𝐴) ∈ {∅, 1o} ↔ ((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1o))
21	18, 20	bitri 278	. . . 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 9439	. . . . . 6 ⊢ (2o ∈ ω → (card‘2o) = 2o)
24	4, 23	ax-mp 5	. . . . 5 ⊢ (card‘2o) = 2o
25	24	eleq2i 2844	. . . 4 ⊢ ((card‘𝐴) ∈ (card‘2o) ↔ (card‘𝐴) ∈ 2o)
26		finnum 9424	. . . . 5 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card)
27		2on 8128	. . . . . 6 ⊢ 2o ∈ On
28		onenon 9425	. . . . . 6 ⊢ (2o ∈ On → 2o ∈ dom card)
29	27, 28	ax-mp 5	. . . . 5 ⊢ 2o ∈ dom card
30		cardsdom2 9464	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 2o ∈ dom card) → ((card‘𝐴) ∈ (card‘2o) ↔ 𝐴 ≺ 2o))
31	26, 29, 30	sylancl 589	. . . 4 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) ∈ (card‘2o) ↔ 𝐴 ≺ 2o))
32	25, 31	bitr3id 288	. . 3 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) ∈ 2o ↔ 𝐴 ≺ 2o))
33		cardnueq0 9440	. . . . 5 ⊢ (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
34	26, 33	syl 17	. . . 4 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
35		cardnn 9439	. . . . . . 7 ⊢ (1o ∈ ω → (card‘1o) = 1o)
36	12, 35	ax-mp 5	. . . . . 6 ⊢ (card‘1o) = 1o
37	36	eqeq2i 2772	. . . . 5 ⊢ ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o)
38		finnum 9424	. . . . . . 7 ⊢ (1o ∈ Fin → 1o ∈ dom card)
39	13, 38	ax-mp 5	. . . . . 6 ⊢ 1o ∈ dom card
40		carden2 9463	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ 1o ∈ dom card) → ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o))
41	26, 39, 40	sylancl 589	. . . . 5 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o))
42	37, 41	bitr3id 288	. . . 4 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) = 1o ↔ 𝐴 ≈ 1o))
43	34, 42	orbi12d 916	. . 3 ⊢ (𝐴 ∈ Fin → (((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1o) ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1o)))
44	22, 32, 43	3bitr3d 312	. 2 ⊢ (𝐴 ∈ Fin → (𝐴 ≺ 2o ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1o)))
45	8, 16, 44	pm5.21nii 383	1 ⊢ (𝐴 ≺ 2o ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1o))

Syntax hints:   ↔ wb 209   ∨ wo 844   = wceq 1539   ∈ wcel 2112   ∩ cin 3860  ∅c0 4228  {cpr 4528   class class class wbr 5037  dom cdm 5529  Oncon0 6175  ‘cfv 6341  ωcom 7586  1_oc1o 8112  2_oc2o 8113   ≈ cen 8538   ≼ cdom 8539   ≺ csdm 8540  Fincfn 8541  cardccrd 9411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5174  ax-nul 5181  ax-pow 5239  ax-pr 5303  ax-un 7466

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3700  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-pss 3880  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4803  df-int 4843  df-br 5038  df-opab 5100  df-mpt 5118  df-tr 5144  df-id 5435  df-eprel 5440  df-po 5448  df-so 5449  df-fr 5488  df-we 5490  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-ord 6178  df-on 6179  df-lim 6180  df-suc 6181  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-om 7587  df-1o 8119  df-2o 8120  df-er 8306  df-en 8542  df-dom 8543  df-sdom 8544  df-fin 8545  df-card 9415

This theorem is referenced by:  fin56  9867  en2top  21700