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Theorem dom0 6813
Description: A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.)
Assertion
Ref Expression
dom0  |-  ( A  ~<_  (/) 
<->  A  =  (/) )

Proof of Theorem dom0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 brdomi 6724 . . 3  |-  ( A  ~<_  (/)  ->  E. x  x : A -1-1-> (/) )
2 f1f 5401 . . . . . 6  |-  ( x : A -1-1-> (/)  ->  x : A --> (/) )
3 f00 5387 . . . . . 6  |-  ( x : A --> (/)  <->  ( x  =  (/)  /\  A  =  (/) ) )
42, 3sylib 121 . . . . 5  |-  ( x : A -1-1-> (/)  ->  (
x  =  (/)  /\  A  =  (/) ) )
54simprd 113 . . . 4  |-  ( x : A -1-1-> (/)  ->  A  =  (/) )
65adantl 275 . . 3  |-  ( ( A  ~<_  (/)  /\  x : A -1-1-> (/) )  ->  A  =  (/) )
71, 6exlimddv 1891 . 2  |-  ( A  ~<_  (/)  ->  A  =  (/) )
8 0ex 4114 . . . 4  |-  (/)  e.  _V
9 domrefg 6742 . . . 4  |-  ( (/)  e.  _V  ->  (/)  ~<_  (/) )
108, 9ax-mp 5 . . 3  |-  (/)  ~<_  (/)
11 breq1 3990 . . 3  |-  ( A  =  (/)  ->  ( A  ~<_  (/) 
<->  (/) 
~<_  (/) ) )
1210, 11mpbiri 167 . 2  |-  ( A  =  (/)  ->  A  ~<_  (/) )
137, 12impbii 125 1  |-  ( A  ~<_  (/) 
<->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1348    e. wcel 2141   _Vcvv 2730   (/)c0 3414   class class class wbr 3987   -->wf 5192   -1-1->wf1 5193    ~<_ cdom 6714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-en 6716  df-dom 6717
This theorem is referenced by: (None)
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