Theorem sdom2en01 9412

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 8394	. . . . 5 ⊢ ω = (On ∩ Fin)
2		inss2 4029	. . . . 5 ⊢ (On ∩ Fin) ⊆ Fin
3	1, 2	eqsstri 3831	. . . 4 ⊢ ω ⊆ Fin
4		2onn 7960	. . . 4 ⊢ 2𝑜 ∈ ω
5	3, 4	sselii 3795	. . 3 ⊢ 2𝑜 ∈ Fin
6		sdomdom 8223	. . 3 ⊢ (𝐴 ≺ 2𝑜 → 𝐴 ≼ 2𝑜)
7		domfi 8423	. . 3 ⊢ ((2𝑜 ∈ Fin ∧ 𝐴 ≼ 2𝑜) → 𝐴 ∈ Fin)
8	5, 6, 7	sylancr 582	. 2 ⊢ (𝐴 ≺ 2𝑜 → 𝐴 ∈ Fin)
9		id 22	. . . 4 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
10		0fin 8430	. . . 4 ⊢ ∅ ∈ Fin
11	9, 10	syl6eqel 2886	. . 3 ⊢ (𝐴 = ∅ → 𝐴 ∈ Fin)
12		1onn 7959	. . . . 5 ⊢ 1𝑜 ∈ ω
13	3, 12	sselii 3795	. . . 4 ⊢ 1𝑜 ∈ Fin
14		enfi 8418	. . . 4 ⊢ (𝐴 ≈ 1𝑜 → (𝐴 ∈ Fin ↔ 1𝑜 ∈ Fin))
15	13, 14	mpbiri 250	. . 3 ⊢ (𝐴 ≈ 1𝑜 → 𝐴 ∈ Fin)
16	11, 15	jaoi 884	. 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 ≈ 1𝑜) → 𝐴 ∈ Fin)
17		df2o3 7813	. . . . . 6 ⊢ 2𝑜 = {∅, 1𝑜}
18	17	eleq2i 2870	. . . . 5 ⊢ ((card‘𝐴) ∈ 2𝑜 ↔ (card‘𝐴) ∈ {∅, 1𝑜})
19		fvex 6424	. . . . . 6 ⊢ (card‘𝐴) ∈ V
20	19	elpr 4391	. . . . 5 ⊢ ((card‘𝐴) ∈ {∅, 1𝑜} ↔ ((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1𝑜))
21	18, 20	bitri 267	. . . 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 9075	. . . . . 6 ⊢ (2𝑜 ∈ ω → (card‘2𝑜) = 2𝑜)
24	4, 23	ax-mp 5	. . . . 5 ⊢ (card‘2𝑜) = 2𝑜
25	24	eleq2i 2870	. . . 4 ⊢ ((card‘𝐴) ∈ (card‘2𝑜) ↔ (card‘𝐴) ∈ 2𝑜)
26		finnum 9060	. . . . 5 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card)
27		2on 7808	. . . . . 6 ⊢ 2𝑜 ∈ On
28		onenon 9061	. . . . . 6 ⊢ (2𝑜 ∈ On → 2𝑜 ∈ dom card)
29	27, 28	ax-mp 5	. . . . 5 ⊢ 2𝑜 ∈ dom card
30		cardsdom2 9100	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 2𝑜 ∈ dom card) → ((card‘𝐴) ∈ (card‘2𝑜) ↔ 𝐴 ≺ 2𝑜))
31	26, 29, 30	sylancl 581	. . . 4 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) ∈ (card‘2𝑜) ↔ 𝐴 ≺ 2𝑜))
32	25, 31	syl5bbr 277	. . 3 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) ∈ 2𝑜 ↔ 𝐴 ≺ 2𝑜))
33		cardnueq0 9076	. . . . 5 ⊢ (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
34	26, 33	syl 17	. . . 4 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
35		cardnn 9075	. . . . . . 7 ⊢ (1𝑜 ∈ ω → (card‘1𝑜) = 1𝑜)
36	12, 35	ax-mp 5	. . . . . 6 ⊢ (card‘1𝑜) = 1𝑜
37	36	eqeq2i 2811	. . . . 5 ⊢ ((card‘𝐴) = (card‘1𝑜) ↔ (card‘𝐴) = 1𝑜)
38		finnum 9060	. . . . . . 7 ⊢ (1𝑜 ∈ Fin → 1𝑜 ∈ dom card)
39	13, 38	ax-mp 5	. . . . . 6 ⊢ 1𝑜 ∈ dom card
40		carden2 9099	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ 1𝑜 ∈ dom card) → ((card‘𝐴) = (card‘1𝑜) ↔ 𝐴 ≈ 1𝑜))
41	26, 39, 40	sylancl 581	. . . . 5 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) = (card‘1𝑜) ↔ 𝐴 ≈ 1𝑜))
42	37, 41	syl5bbr 277	. . . 4 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) = 1𝑜 ↔ 𝐴 ≈ 1𝑜))
43	34, 42	orbi12d 943	. . 3 ⊢ (𝐴 ∈ Fin → (((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1𝑜) ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1𝑜)))
44	22, 32, 43	3bitr3d 301	. 2 ⊢ (𝐴 ∈ Fin → (𝐴 ≺ 2𝑜 ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1𝑜)))
45	8, 16, 44	pm5.21nii 370	1 ⊢ (𝐴 ≺ 2𝑜 ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1𝑜))

Syntax hints:   ↔ wb 198   ∨ wo 874   = wceq 1653   ∈ wcel 2157   ∩ cin 3768  ∅c0 4115  {cpr 4370   class class class wbr 4843  dom cdm 5312  Oncon0 5941  ‘cfv 6101  ωcom 7299  1_𝑜c1o 7792  2_𝑜c2o 7793   ≈ cen 8192   ≼ cdom 8193   ≺ csdm 8194  Fincfn 8195  cardccrd 9047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-om 7300  df-1o 7799  df-2o 7800  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-card 9051

This theorem is referenced by:  fin56  9503  en2top  21118