Theorem sdom2en01 10218

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 9145	. . . . 5 ⊢ ω = (On ∩ Fin)
2		inss2 4179	. . . . 5 ⊢ (On ∩ Fin) ⊆ Fin
3	1, 2	eqsstri 3969	. . . 4 ⊢ ω ⊆ Fin
4		2onn 8572	. . . 4 ⊢ 2o ∈ ω
5	3, 4	sselii 3919	. . 3 ⊢ 2o ∈ Fin
6		sdomdom 8921	. . 3 ⊢ (𝐴 ≺ 2o → 𝐴 ≼ 2o)
7		domfi 9117	. . 3 ⊢ ((2o ∈ Fin ∧ 𝐴 ≼ 2o) → 𝐴 ∈ Fin)
8	5, 6, 7	sylancr 588	. 2 ⊢ (𝐴 ≺ 2o → 𝐴 ∈ Fin)
9		id 22	. . . 4 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
10		0fi 8983	. . . 4 ⊢ ∅ ∈ Fin
11	9, 10	eqeltrdi 2845	. . 3 ⊢ (𝐴 = ∅ → 𝐴 ∈ Fin)
12		1onn 8570	. . . . 5 ⊢ 1o ∈ ω
13	3, 12	sselii 3919	. . . 4 ⊢ 1o ∈ Fin
14		enfi 9115	. . . 4 ⊢ (𝐴 ≈ 1o → (𝐴 ∈ Fin ↔ 1o ∈ Fin))
15	13, 14	mpbiri 258	. . 3 ⊢ (𝐴 ≈ 1o → 𝐴 ∈ Fin)
16	11, 15	jaoi 858	. 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 ≈ 1o) → 𝐴 ∈ Fin)
17		df2o3 8407	. . . . . 6 ⊢ 2o = {∅, 1o}
18	17	eleq2i 2829	. . . . 5 ⊢ ((card‘𝐴) ∈ 2o ↔ (card‘𝐴) ∈ {∅, 1o})
19		fvex 6848	. . . . . 6 ⊢ (card‘𝐴) ∈ V
20	19	elpr 4593	. . . . 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 9881	. . . . . 6 ⊢ (2o ∈ ω → (card‘2o) = 2o)
24	4, 23	ax-mp 5	. . . . 5 ⊢ (card‘2o) = 2o
25	24	eleq2i 2829	. . . 4 ⊢ ((card‘𝐴) ∈ (card‘2o) ↔ (card‘𝐴) ∈ 2o)
26		finnum 9866	. . . . 5 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card)
27		2on 8412	. . . . . 6 ⊢ 2o ∈ On
28		onenon 9867	. . . . . 6 ⊢ (2o ∈ On → 2o ∈ dom card)
29	27, 28	ax-mp 5	. . . . 5 ⊢ 2o ∈ dom card
30		cardsdom2 9906	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 2o ∈ dom card) → ((card‘𝐴) ∈ (card‘2o) ↔ 𝐴 ≺ 2o))
31	26, 29, 30	sylancl 587	. . . 4 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) ∈ (card‘2o) ↔ 𝐴 ≺ 2o))
32	25, 31	bitr3id 285	. . 3 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) ∈ 2o ↔ 𝐴 ≺ 2o))
33		cardnueq0 9882	. . . . 5 ⊢ (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
34	26, 33	syl 17	. . . 4 ⊢ (𝐴 ∈ Fin → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
35		cardnn 9881	. . . . . . 7 ⊢ (1o ∈ ω → (card‘1o) = 1o)
36	12, 35	ax-mp 5	. . . . . 6 ⊢ (card‘1o) = 1o
37	36	eqeq2i 2750	. . . . 5 ⊢ ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o)
38		finnum 9866	. . . . . . 7 ⊢ (1o ∈ Fin → 1o ∈ dom card)
39	13, 38	ax-mp 5	. . . . . 6 ⊢ 1o ∈ dom card
40		carden2 9905	. . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ 1o ∈ dom card) → ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o))
41	26, 39, 40	sylancl 587	. . . . 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 919	. . 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 848   = wceq 1542   ∈ wcel 2114   ∩ cin 3889  ∅c0 4274  {cpr 4570   class class class wbr 5086  dom cdm 5625  Oncon0 6318  ‘cfv 6493  ωcom 7811  1_oc1o 8392  2_oc2o 8393   ≈ cen 8884   ≼ cdom 8885   ≺ csdm 8886  Fincfn 8887  cardccrd 9853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-om 7812  df-1o 8399  df-2o 8400  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-card 9857

This theorem is referenced by:  fin56  10309  en2top  22963