Theorem sdom2en01 9124

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 8152	. . . . 5 ⊢ ω = (On ∩ Fin)
2		inss2 3834	. . . . 5 ⊢ (On ∩ Fin) ⊆ Fin
3	1, 2	eqsstri 3635	. . . 4 ⊢ ω ⊆ Fin
4		2onn 7720	. . . 4 ⊢ 2𝑜 ∈ ω
5	3, 4	sselii 3600	. . 3 ⊢ 2𝑜 ∈ Fin
6		sdomdom 7983	. . 3 ⊢ (𝐴 ≺ 2𝑜 → 𝐴 ≼ 2𝑜)
7		domfi 8181	. . 3 ⊢ ((2𝑜 ∈ Fin ∧ 𝐴 ≼ 2𝑜) → 𝐴 ∈ Fin)
8	5, 6, 7	sylancr 695	. 2 ⊢ (𝐴 ≺ 2𝑜 → 𝐴 ∈ Fin)
9		id 22	. . . 4 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
10		0fin 8188	. . . 4 ⊢ ∅ ∈ Fin
11	9, 10	syl6eqel 2709	. . 3 ⊢ (𝐴 = ∅ → 𝐴 ∈ Fin)
12		1onn 7719	. . . . 5 ⊢ 1𝑜 ∈ ω
13	3, 12	sselii 3600	. . . 4 ⊢ 1𝑜 ∈ Fin
14		enfi 8176	. . . 4 ⊢ (𝐴 ≈ 1𝑜 → (𝐴 ∈ Fin ↔ 1𝑜 ∈ Fin))
15	13, 14	mpbiri 248	. . 3 ⊢ (𝐴 ≈ 1𝑜 → 𝐴 ∈ Fin)
16	11, 15	jaoi 394	. 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 ≈ 1𝑜) → 𝐴 ∈ Fin)
17		df2o3 7573	. . . . . 6 ⊢ 2𝑜 = {∅, 1𝑜}
18	17	eleq2i 2693	. . . . 5 ⊢ ((card‘𝐴) ∈ 2𝑜 ↔ (card‘𝐴) ∈ {∅, 1𝑜})
19		fvex 6201	. . . . . 6 ⊢ (card‘𝐴) ∈ V
20	19	elpr 4198	. . . . 5 ⊢ ((card‘𝐴) ∈ {∅, 1𝑜} ↔ ((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1𝑜))
21	18, 20	bitri 264	. . . 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 8789	. . . . . 6 ⊢ (2𝑜 ∈ ω → (card‘2𝑜) = 2𝑜)
24	4, 23	ax-mp 5	. . . . 5 ⊢ (card‘2𝑜) = 2𝑜
25	24	eleq2i 2693	. . . 4 ⊢ ((card‘𝐴) ∈ (card‘2𝑜) ↔ (card‘𝐴) ∈ 2𝑜)
26		finnum 8774	. . . . 5 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card)
27		2on 7568	. . . . . 6 ⊢ 2𝑜 ∈ On
28		onenon 8775	. . . . . 6 ⊢ (2𝑜 ∈ On → 2𝑜 ∈ dom card)
29	27, 28	ax-mp 5	. . . . 5 ⊢ 2𝑜 ∈ dom card
30		cardsdom2 8814	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 2𝑜 ∈ dom card) → ((card‘𝐴) ∈ (card‘2𝑜) ↔ 𝐴 ≺ 2𝑜))
31	26, 29, 30	sylancl 694	. . . 4 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) ∈ (card‘2𝑜) ↔ 𝐴 ≺ 2𝑜))
32	25, 31	syl5bbr 274	. . 3 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) ∈ 2𝑜 ↔ 𝐴 ≺ 2𝑜))
33		cardnueq0 8790	. . . . 5 ⊢ (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
34	26, 33	syl 17	. . . 4 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
35		cardnn 8789	. . . . . . 7 ⊢ (1𝑜 ∈ ω → (card‘1𝑜) = 1𝑜)
36	12, 35	ax-mp 5	. . . . . 6 ⊢ (card‘1𝑜) = 1𝑜
37	36	eqeq2i 2634	. . . . 5 ⊢ ((card‘𝐴) = (card‘1𝑜) ↔ (card‘𝐴) = 1𝑜)
38		finnum 8774	. . . . . . 7 ⊢ (1𝑜 ∈ Fin → 1𝑜 ∈ dom card)
39	13, 38	ax-mp 5	. . . . . 6 ⊢ 1𝑜 ∈ dom card
40		carden2 8813	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ 1𝑜 ∈ dom card) → ((card‘𝐴) = (card‘1𝑜) ↔ 𝐴 ≈ 1𝑜))
41	26, 39, 40	sylancl 694	. . . . 5 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) = (card‘1𝑜) ↔ 𝐴 ≈ 1𝑜))
42	37, 41	syl5bbr 274	. . . 4 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) = 1𝑜 ↔ 𝐴 ≈ 1𝑜))
43	34, 42	orbi12d 746	. . 3 ⊢ (𝐴 ∈ Fin → (((card‘𝐴) = ∅ ∨ (card‘𝐴) = 1𝑜) ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1𝑜)))
44	22, 32, 43	3bitr3d 298	. 2 ⊢ (𝐴 ∈ Fin → (𝐴 ≺ 2𝑜 ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1𝑜)))
45	8, 16, 44	pm5.21nii 368	1 ⊢ (𝐴 ≺ 2𝑜 ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1𝑜))

Syntax hints:   ↔ wb 196   ∨ wo 383   = wceq 1483   ∈ wcel 1990   ∩ cin 3573  ∅c0 3915  {cpr 4179   class class class wbr 4653  dom cdm 5114  Oncon0 5723  ‘cfv 5888  ωcom 7065  1_𝑜c1o 7553  2_𝑜c2o 7554   ≈ cen 7952   ≼ cdom 7953   ≺ csdm 7954  Fincfn 7955  cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765

This theorem is referenced by:  fin56  9215  en2top  20789