Theorem sdom2en01 10058

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 9014	. . . . 5 ⊢ ω = (On ∩ Fin)
2		inss2 4163	. . . . 5 ⊢ (On ∩ Fin) ⊆ Fin
3	1, 2	eqsstri 3955	. . . 4 ⊢ ω ⊆ Fin
4		2onn 8472	. . . 4 ⊢ 2o ∈ ω
5	3, 4	sselii 3918	. . 3 ⊢ 2o ∈ Fin
6		sdomdom 8768	. . 3 ⊢ (𝐴 ≺ 2o → 𝐴 ≼ 2o)
7		domfi 8975	. . 3 ⊢ ((2o ∈ Fin ∧ 𝐴 ≼ 2o) → 𝐴 ∈ Fin)
8	5, 6, 7	sylancr 587	. 2 ⊢ (𝐴 ≺ 2o → 𝐴 ∈ Fin)
9		id 22	. . . 4 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
10		0fin 8954	. . . 4 ⊢ ∅ ∈ Fin
11	9, 10	eqeltrdi 2847	. . 3 ⊢ (𝐴 = ∅ → 𝐴 ∈ Fin)
12		1onn 8470	. . . . 5 ⊢ 1o ∈ ω
13	3, 12	sselii 3918	. . . 4 ⊢ 1o ∈ Fin
14		enfi 8973	. . . 4 ⊢ (𝐴 ≈ 1o → (𝐴 ∈ Fin ↔ 1o ∈ Fin))
15	13, 14	mpbiri 257	. . 3 ⊢ (𝐴 ≈ 1o → 𝐴 ∈ Fin)
16	11, 15	jaoi 854	. 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 ≈ 1o) → 𝐴 ∈ Fin)
17		df2o3 8305	. . . . . 6 ⊢ 2o = {∅, 1o}
18	17	eleq2i 2830	. . . . 5 ⊢ ((card‘𝐴) ∈ 2o ↔ (card‘𝐴) ∈ {∅, 1o})
19		fvex 6787	. . . . . 6 ⊢ (card‘𝐴) ∈ V
20	19	elpr 4584	. . . . 5 ⊢ ((card‘𝐴) ∈ {∅, 1o} ↔ ((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1o))
21	18, 20	bitri 274	. . . 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 9721	. . . . . 6 ⊢ (2o ∈ ω → (card‘2o) = 2o)
24	4, 23	ax-mp 5	. . . . 5 ⊢ (card‘2o) = 2o
25	24	eleq2i 2830	. . . 4 ⊢ ((card‘𝐴) ∈ (card‘2o) ↔ (card‘𝐴) ∈ 2o)
26		finnum 9706	. . . . 5 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card)
27		2on 8311	. . . . . 6 ⊢ 2o ∈ On
28		onenon 9707	. . . . . 6 ⊢ (2o ∈ On → 2o ∈ dom card)
29	27, 28	ax-mp 5	. . . . 5 ⊢ 2o ∈ dom card
30		cardsdom2 9746	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 2o ∈ dom card) → ((card‘𝐴) ∈ (card‘2o) ↔ 𝐴 ≺ 2o))
31	26, 29, 30	sylancl 586	. . . 4 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) ∈ (card‘2o) ↔ 𝐴 ≺ 2o))
32	25, 31	bitr3id 285	. . 3 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) ∈ 2o ↔ 𝐴 ≺ 2o))
33		cardnueq0 9722	. . . . 5 ⊢ (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
34	26, 33	syl 17	. . . 4 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
35		cardnn 9721	. . . . . . 7 ⊢ (1o ∈ ω → (card‘1o) = 1o)
36	12, 35	ax-mp 5	. . . . . 6 ⊢ (card‘1o) = 1o
37	36	eqeq2i 2751	. . . . 5 ⊢ ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o)
38		finnum 9706	. . . . . . 7 ⊢ (1o ∈ Fin → 1o ∈ dom card)
39	13, 38	ax-mp 5	. . . . . 6 ⊢ 1o ∈ dom card
40		carden2 9745	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ 1o ∈ dom card) → ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o))
41	26, 39, 40	sylancl 586	. . . . 5 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o))
42	37, 41	bitr3id 285	. . . 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 309	. 2 ⊢ (𝐴 ∈ Fin → (𝐴 ≺ 2o ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1o)))
45	8, 16, 44	pm5.21nii 380	1 ⊢ (𝐴 ≺ 2o ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1o))

Syntax hints:   ↔ wb 205   ∨ wo 844   = wceq 1539   ∈ wcel 2106   ∩ cin 3886  ∅c0 4256  {cpr 4563   class class class wbr 5074  dom cdm 5589  Oncon0 6266  ‘cfv 6433  ωcom 7712  1_oc1o 8290  2_oc2o 8291   ≈ cen 8730   ≼ cdom 8731   ≺ csdm 8732  Fincfn 8733  cardccrd 9693

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-1o 8297  df-2o 8298  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-card 9697

This theorem is referenced by:  fin56  10149  en2top  22135