MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sdom1 Unicode version

Theorem sdom1 7078
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
StepHypRef Expression
1 domnsym 7003 . . . . 5  |-  ( 1o  ~<_  A  ->  -.  A  ~<  1o )
21con2i 112 . . . 4  |-  ( A 
~<  1o  ->  -.  1o  ~<_  A )
3 0sdom1dom 7076 . . . 4  |-  ( (/)  ~<  A 
<->  1o  ~<_  A )
42, 3sylnibr 296 . . 3  |-  ( A 
~<  1o  ->  -.  (/)  ~<  A )
5 relsdom 6886 . . . . 5  |-  Rel  ~<
65brrelexi 4745 . . . 4  |-  ( A 
~<  1o  ->  A  e.  _V )
7 0sdomg 7006 . . . . 5  |-  ( A  e.  _V  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
87necon2bbid 2517 . . . 4  |-  ( A  e.  _V  ->  ( A  =  (/)  <->  -.  (/)  ~<  A ) )
96, 8syl 15 . . 3  |-  ( A 
~<  1o  ->  ( A  =  (/)  <->  -.  (/)  ~<  A ) )
104, 9mpbird 223 . 2  |-  ( A 
~<  1o  ->  A  =  (/) )
11 1n0 6510 . . . 4  |-  1o  =/=  (/)
12 1on 6502 . . . . . 6  |-  1o  e.  On
1312elexi 2810 . . . . 5  |-  1o  e.  _V
14130sdom 7008 . . . 4  |-  ( (/)  ~<  1o 
<->  1o  =/=  (/) )
1511, 14mpbir 200 . . 3  |-  (/)  ~<  1o
16 breq1 4042 . . 3  |-  ( A  =  (/)  ->  ( A 
~<  1o  <->  (/)  ~<  1o )
)
1715, 16mpbiri 224 . 2  |-  ( A  =  (/)  ->  A  ~<  1o )
1810, 17impbii 180 1  |-  ( A 
~<  1o  <->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801   (/)c0 3468   class class class wbr 4039   Oncon0 4408   1oc1o 6488    ~<_ cdom 6877    ~< csdm 6878
This theorem is referenced by:  modom  7079  frgpcyg  16543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882
  Copyright terms: Public domain W3C validator