Theorem sdom1 7790

Description: A set has less than one member iff it is empty. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Assertion

Ref	Expression
sdom1	|- ( A ~< 1o <-> A = (/) )

Proof of Theorem sdom1

Step	Hyp	Ref	Expression
1		domnsym 7716	. . . . 5 |- ( 1o ~<_ A -> -. A ~< 1o )
2	1	con2i 124	. . . 4 |- ( A ~< 1o -> -. 1o ~<_ A )
3		0sdom1dom 7788	. . . 4 |- ( (/) ~< A <-> 1o ~<_ A )
4	2, 3	sylnibr 312	. . 3 |- ( A ~< 1o -> -. (/) ~< A )
5		relsdom 7594	. . . . 5 |- Rel ~<
6	5	brrelexi 4880	. . . 4 |- ( A ~< 1o -> A e. _V )
7		0sdomg 7719	. . . . 5 |- ( A e. _V -> ( (/) ~< A <-> A =/= (/) ) )
8	7	necon2bbid 2686	. . . 4 |- ( A e. _V -> ( A = (/) <-> -. (/) ~< A ) )
9	6, 8	syl 17	. . 3 |- ( A ~< 1o -> ( A = (/) <-> -. (/) ~< A ) )
10	4, 9	mpbird 240	. 2 |- ( A ~< 1o -> A = (/) )
11		1n0 7215	. . . 4 |- 1o =/= (/)
12		1on 7207	. . . . . 6 |- 1o e. On
13	12	elexi 3041	. . . . 5 |- 1o e. _V
14	13	0sdom 7721	. . . 4 |- ( (/) ~< 1o <-> 1o =/= (/) )
15	11, 14	mpbir 214	. . 3 |- (/) ~< 1o
16		breq1 4398	. . 3 |- ( A = (/) -> ( A ~< 1o <-> (/) ~< 1o ) )
17	15, 16	mpbiri 241	. 2 |- ( A = (/) -> A ~< 1o )
18	10, 17	impbii 192	1 |- ( A ~< 1o <-> A = (/) )

Syntax hints:   -. wn 3   <-> wb 189   = wceq 1452   e. wcel 1904   =/= wne 2641   _Vcvv 3031   (/)c0 3722   class class class wbr 4395   Oncon0 5430   1oc1o 7193   ~<_ cdom 7585   ~< csdm 7586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-om 6712  df-1o 7200  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590

This theorem is referenced by:  modom  7791  frgpcyg  19221