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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
StepHypRef Expression
1 domnsym 7636 . . . . 5  |-  ( 1o  ~<_  A  ->  -.  A  ~<  1o )
21con2i 120 . . . 4  |-  ( A 
~<  1o  ->  -.  1o  ~<_  A )
3 0sdom1dom 7710 . . . 4  |-  ( (/)  ~<  A 
<->  1o  ~<_  A )
42, 3sylnibr 303 . . 3  |-  ( A 
~<  1o  ->  -.  (/)  ~<  A )
5 relsdom 7516 . . . . 5  |-  Rel  ~<
65brrelexi 5029 . . . 4  |-  ( A 
~<  1o  ->  A  e.  _V )
7 0sdomg 7639 . . . . 5  |-  ( A  e.  _V  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
87necon2bbid 2710 . . . 4  |-  ( A  e.  _V  ->  ( A  =  (/)  <->  -.  (/)  ~<  A ) )
96, 8syl 16 . . 3  |-  ( A 
~<  1o  ->  ( A  =  (/)  <->  -.  (/)  ~<  A ) )
104, 9mpbird 232 . 2  |-  ( A 
~<  1o  ->  A  =  (/) )
11 1n0 7137 . . . 4  |-  1o  =/=  (/)
12 1on 7129 . . . . . 6  |-  1o  e.  On
1312elexi 3116 . . . . 5  |-  1o  e.  _V
14130sdom 7641 . . . 4  |-  ( (/)  ~<  1o 
<->  1o  =/=  (/) )
1511, 14mpbir 209 . . 3  |-  (/)  ~<  1o
16 breq1 4442 . . 3  |-  ( A  =  (/)  ->  ( A 
~<  1o  <->  (/)  ~<  1o )
)
1715, 16mpbiri 233 . 2  |-  ( A  =  (/)  ->  A  ~<  1o )
1810, 17impbii 188 1  |-  ( A 
~<  1o  <->  A  =  (/) )
Colors of variables: wff setvar class
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
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