Theorem sdom1 7712

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 7636	. . . . 5 |- ( 1o ~<_ A -> -. A ~< 1o )
2	1	con2i 120	. . . 4 |- ( A ~< 1o -> -. 1o ~<_ A )
3		0sdom1dom 7710	. . . 4 |- ( (/) ~< A <-> 1o ~<_ A )
4	2, 3	sylnibr 303	. . 3 |- ( A ~< 1o -> -. (/) ~< A )
5		relsdom 7516	. . . . 5 |- Rel ~<
6	5	brrelexi 5029	. . . 4 |- ( A ~< 1o -> A e. _V )
7		0sdomg 7639	. . . . 5 |- ( A e. _V -> ( (/) ~< A <-> A =/= (/) ) )
8	7	necon2bbid 2710	. . . 4 |- ( A e. _V -> ( A = (/) <-> -. (/) ~< A ) )
9	6, 8	syl 16	. . 3 |- ( A ~< 1o -> ( A = (/) <-> -. (/) ~< A ) )
10	4, 9	mpbird 232	. 2 |- ( A ~< 1o -> A = (/) )
11		1n0 7137	. . . 4 |- 1o =/= (/)
12		1on 7129	. . . . . 6 |- 1o e. On
13	12	elexi 3116	. . . . 5 |- 1o e. _V
14	13	0sdom 7641	. . . 4 |- ( (/) ~< 1o <-> 1o =/= (/) )
15	11, 14	mpbir 209	. . 3 |- (/) ~< 1o
16		breq1 4442	. . 3 |- ( A = (/) -> ( A ~< 1o <-> (/) ~< 1o ) )
17	15, 16	mpbiri 233	. 2 |- ( A = (/) -> A ~< 1o )
18	10, 17	impbii 188	1 |- ( A ~< 1o <-> A = (/) )

Syntax hints:   -. wn 3   <-> wb 184   = wceq 1398   e. wcel 1823   =/= wne 2649   _Vcvv 3106   (/)c0 3783   class class class wbr 4439   Oncon0 4867   1oc1o 7115   ~<_ cdom 7507   ~< csdm 7508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-1o 7122  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512

This theorem is referenced by:  modom  7713  frgpcyg  18785