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 |- om = ( On i^i Fin )
2		inss2 3834	. . . . 5 |- ( On i^i Fin ) C_ Fin
3	1, 2	eqsstri 3635	. . . 4 |- om C_ Fin
4		2onn 7720	. . . 4 |- 2o e. om
5	3, 4	sselii 3600	. . 3 |- 2o e. Fin
6		sdomdom 7983	. . 3 |- ( A ~< 2o -> A ~<_ 2o )
7		domfi 8181	. . 3 |- ( ( 2o e. Fin /\ A ~<_ 2o ) -> A e. Fin )
8	5, 6, 7	sylancr 695	. 2 |- ( A ~< 2o -> A e. Fin )
9		id 22	. . . 4 |- ( A = (/) -> A = (/) )
10		0fin 8188	. . . 4 |- (/) e. Fin
11	9, 10	syl6eqel 2709	. . 3 |- ( A = (/) -> A e. Fin )
12		1onn 7719	. . . . 5 |- 1o e. om
13	3, 12	sselii 3600	. . . 4 |- 1o e. Fin
14		enfi 8176	. . . 4 |- ( A ~~ 1o -> ( A e. Fin <-> 1o e. Fin ) )
15	13, 14	mpbiri 248	. . 3 |- ( A ~~ 1o -> A e. Fin )
16	11, 15	jaoi 394	. 2 |- ( ( A = (/) \/ A ~~ 1o ) -> A e. Fin )
17		df2o3 7573	. . . . . 6 |- 2o = { (/) , 1o }
18	17	eleq2i 2693	. . . . 5 |- ( ( card ` A ) e. 2o <-> ( card ` A ) e. { (/) , 1o } )
19		fvex 6201	. . . . . 6 |- ( card ` A ) e. _V
20	19	elpr 4198	. . . . 5 |- ( ( card ` A ) e. { (/) , 1o } <-> ( ( card ` A ) = (/) \/ ( card ` A ) = 1o ) )
21	18, 20	bitri 264	. . . 4 |- ( ( card ` A ) e. 2o <-> ( ( card ` A ) = (/) \/ ( card ` A ) = 1o ) )
22	21	a1i 11	. . 3 |- ( A e. Fin -> ( ( card ` A ) e. 2o <-> ( ( card ` A ) = (/) \/ ( card ` A ) = 1o ) ) )
23		cardnn 8789	. . . . . 6 |- ( 2o e. om -> ( card ` 2o ) = 2o )
24	4, 23	ax-mp 5	. . . . 5 |- ( card ` 2o ) = 2o
25	24	eleq2i 2693	. . . 4 |- ( ( card ` A ) e. ( card ` 2o ) <-> ( card ` A ) e. 2o )
26		finnum 8774	. . . . 5 |- ( A e. Fin -> A e. dom card )
27		2on 7568	. . . . . 6 |- 2o e. On
28		onenon 8775	. . . . . 6 |- ( 2o e. On -> 2o e. dom card )
29	27, 28	ax-mp 5	. . . . 5 |- 2o e. dom card
30		cardsdom2 8814	. . . . 5 |- ( ( A e. dom card /\ 2o e. dom card ) -> ( ( card ` A ) e. ( card ` 2o ) <-> A ~< 2o ) )
31	26, 29, 30	sylancl 694	. . . 4 |- ( A e. Fin -> ( ( card ` A ) e. ( card ` 2o ) <-> A ~< 2o ) )
32	25, 31	syl5bbr 274	. . 3 |- ( A e. Fin -> ( ( card ` A ) e. 2o <-> A ~< 2o ) )
33		cardnueq0 8790	. . . . 5 |- ( A e. dom card -> ( ( card ` A ) = (/) <-> A = (/) ) )
34	26, 33	syl 17	. . . 4 |- ( A e. Fin -> ( ( card ` A ) = (/) <-> A = (/) ) )
35		cardnn 8789	. . . . . . 7 |- ( 1o e. om -> ( 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 e. Fin -> 1o e. dom card )
39	13, 38	ax-mp 5	. . . . . 6 |- 1o e. dom card
40		carden2 8813	. . . . . 6 |- ( ( A e. dom card /\ 1o e. dom card ) -> ( ( card ` A ) = ( card ` 1o ) <-> A ~~ 1o ) )
41	26, 39, 40	sylancl 694	. . . . 5 |- ( A e. Fin -> ( ( card ` A ) = ( card ` 1o ) <-> A ~~ 1o ) )
42	37, 41	syl5bbr 274	. . . 4 |- ( A e. Fin -> ( ( card ` A ) = 1o <-> A ~~ 1o ) )
43	34, 42	orbi12d 746	. . 3 |- ( A e. Fin -> ( ( ( card ` A ) = (/) \/ ( card ` A ) = 1o ) <-> ( A = (/) \/ A ~~ 1o ) ) )
44	22, 32, 43	3bitr3d 298	. 2 |- ( A e. 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   e. wcel 1990   i^i cin 3573   (/)c0 3915   {cpr 4179   class class class wbr 4653   dom cdm 5114   Oncon0 5723   ` cfv 5888   omcom 7065   1oc1o 7553   2oc2o 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