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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
StepHypRef Expression
1 domnsym 7716 . . . . 5  |-  ( 1o  ~<_  A  ->  -.  A  ~<  1o )
21con2i 124 . . . 4  |-  ( A 
~<  1o  ->  -.  1o  ~<_  A )
3 0sdom1dom 7788 . . . 4  |-  ( (/)  ~<  A 
<->  1o  ~<_  A )
42, 3sylnibr 312 . . 3  |-  ( A 
~<  1o  ->  -.  (/)  ~<  A )
5 relsdom 7594 . . . . 5  |-  Rel  ~<
65brrelexi 4880 . . . 4  |-  ( A 
~<  1o  ->  A  e.  _V )
7 0sdomg 7719 . . . . 5  |-  ( A  e.  _V  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
87necon2bbid 2686 . . . 4  |-  ( A  e.  _V  ->  ( A  =  (/)  <->  -.  (/)  ~<  A ) )
96, 8syl 17 . . 3  |-  ( A 
~<  1o  ->  ( A  =  (/)  <->  -.  (/)  ~<  A ) )
104, 9mpbird 240 . 2  |-  ( A 
~<  1o  ->  A  =  (/) )
11 1n0 7215 . . . 4  |-  1o  =/=  (/)
12 1on 7207 . . . . . 6  |-  1o  e.  On
1312elexi 3041 . . . . 5  |-  1o  e.  _V
14130sdom 7721 . . . 4  |-  ( (/)  ~<  1o 
<->  1o  =/=  (/) )
1511, 14mpbir 214 . . 3  |-  (/)  ~<  1o
16 breq1 4398 . . 3  |-  ( A  =  (/)  ->  ( A 
~<  1o  <->  (/)  ~<  1o )
)
1715, 16mpbiri 241 . 2  |-  ( A  =  (/)  ->  A  ~<  1o )
1810, 17impbii 192 1  |-  ( A 
~<  1o  <->  A  =  (/) )
Colors of variables: wff setvar class
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
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