Theorem sdom2en01 10316

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 9240	. . . . 5 ⊢ ω = (On ∩ Fin)
2		inss2 4213	. . . . 5 ⊢ (On ∩ Fin) ⊆ Fin
3	1, 2	eqsstri 4005	. . . 4 ⊢ ω ⊆ Fin
4		2onn 8654	. . . 4 ⊢ 2o ∈ ω
5	3, 4	sselii 3955	. . 3 ⊢ 2o ∈ Fin
6		sdomdom 8994	. . 3 ⊢ (𝐴 ≺ 2o → 𝐴 ≼ 2o)
7		domfi 9203	. . 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		0fi 9056	. . . 4 ⊢ ∅ ∈ Fin
11	9, 10	eqeltrdi 2842	. . 3 ⊢ (𝐴 = ∅ → 𝐴 ∈ Fin)
12		1onn 8652	. . . . 5 ⊢ 1o ∈ ω
13	3, 12	sselii 3955	. . . 4 ⊢ 1o ∈ Fin
14		enfi 9201	. . . 4 ⊢ (𝐴 ≈ 1o → (𝐴 ∈ Fin ↔ 1o ∈ Fin))
15	13, 14	mpbiri 258	. . 3 ⊢ (𝐴 ≈ 1o → 𝐴 ∈ Fin)
16	11, 15	jaoi 857	. 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 ≈ 1o) → 𝐴 ∈ Fin)
17		df2o3 8488	. . . . . 6 ⊢ 2o = {∅, 1o}
18	17	eleq2i 2826	. . . . 5 ⊢ ((card‘𝐴) ∈ 2o ↔ (card‘𝐴) ∈ {∅, 1o})
19		fvex 6889	. . . . . 6 ⊢ (card‘𝐴) ∈ V
20	19	elpr 4626	. . . . 5 ⊢ ((card‘𝐴) ∈ {∅, 1o} ↔ ((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1o))
21	18, 20	bitri 275	. . . 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 9977	. . . . . 6 ⊢ (2o ∈ ω → (card‘2o) = 2o)
24	4, 23	ax-mp 5	. . . . 5 ⊢ (card‘2o) = 2o
25	24	eleq2i 2826	. . . 4 ⊢ ((card‘𝐴) ∈ (card‘2o) ↔ (card‘𝐴) ∈ 2o)
26		finnum 9962	. . . . 5 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card)
27		2on 8494	. . . . . 6 ⊢ 2o ∈ On
28		onenon 9963	. . . . . 6 ⊢ (2o ∈ On → 2o ∈ dom card)
29	27, 28	ax-mp 5	. . . . 5 ⊢ 2o ∈ dom card
30		cardsdom2 10002	. . . . 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 9978	. . . . 5 ⊢ (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
34	26, 33	syl 17	. . . 4 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
35		cardnn 9977	. . . . . . 7 ⊢ (1o ∈ ω → (card‘1o) = 1o)
36	12, 35	ax-mp 5	. . . . . 6 ⊢ (card‘1o) = 1o
37	36	eqeq2i 2748	. . . . 5 ⊢ ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o)
38		finnum 9962	. . . . . . 7 ⊢ (1o ∈ Fin → 1o ∈ dom card)
39	13, 38	ax-mp 5	. . . . . 6 ⊢ 1o ∈ dom card
40		carden2 10001	. . . . . 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 918	. . 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 378	1 ⊢ (𝐴 ≺ 2o ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1o))

Syntax hints:   ↔ wb 206   ∨ wo 847   = wceq 1540   ∈ wcel 2108   ∩ cin 3925  ∅c0 4308  {cpr 4603   class class class wbr 5119  dom cdm 5654  Oncon0 6352  ‘cfv 6531  ωcom 7861  1_oc1o 8473  2_oc2o 8474   ≈ cen 8956   ≼ cdom 8957   ≺ csdm 8958  Fincfn 8959  cardccrd 9949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-om 7862  df-1o 8480  df-2o 8481  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-card 9953

This theorem is referenced by:  fin56  10407  en2top  22923