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Theorem sdom1 7062
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 6987 . . . . 5  |-  ( 1o  ~<_  A  ->  -.  A  ~<  1o )
21con2i 112 . . . 4  |-  ( A 
~<  1o  ->  -.  1o  ~<_  A )
3 0sdom1dom 7060 . . . 4  |-  ( (/)  ~<  A 
<->  1o  ~<_  A )
42, 3sylnibr 296 . . 3  |-  ( A 
~<  1o  ->  -.  (/)  ~<  A )
5 relsdom 6870 . . . . 5  |-  Rel  ~<
65brrelexi 4729 . . . 4  |-  ( A 
~<  1o  ->  A  e.  _V )
7 0sdomg 6990 . . . . 5  |-  ( A  e.  _V  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
87necon2bbid 2504 . . . 4  |-  ( A  e.  _V  ->  ( A  =  (/)  <->  -.  (/)  ~<  A ) )
96, 8syl 15 . . 3  |-  ( A 
~<  1o  ->  ( A  =  (/)  <->  -.  (/)  ~<  A ) )
104, 9mpbird 223 . 2  |-  ( A 
~<  1o  ->  A  =  (/) )
11 1n0 6494 . . . 4  |-  1o  =/=  (/)
12 1on 6486 . . . . . 6  |-  1o  e.  On
1312elexi 2797 . . . . 5  |-  1o  e.  _V
14130sdom 6992 . . . 4  |-  ( (/)  ~<  1o 
<->  1o  =/=  (/) )
1511, 14mpbir 200 . . 3  |-  (/)  ~<  1o
16 breq1 4026 . . 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 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   (/)c0 3455   class class class wbr 4023   Oncon0 4392   1oc1o 6472    ~<_ cdom 6861    ~< csdm 6862
This theorem is referenced by:  modom  7063  frgpcyg  16527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866
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