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Theorem dom0 6885
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 6794 . . 3  |-  ( A  ~<_  (/)  ->  E. x  x : A -1-1-> (/) )
2 f1f 5451 . . . . . 6  |-  ( x : A -1-1-> (/)  ->  x : A --> (/) )
3 f00 5437 . . . . . 6  |-  ( x : A --> (/)  <->  ( x  =  (/)  /\  A  =  (/) ) )
42, 3sylib 122 . . . . 5  |-  ( x : A -1-1-> (/)  ->  (
x  =  (/)  /\  A  =  (/) ) )
54simprd 114 . . . 4  |-  ( x : A -1-1-> (/)  ->  A  =  (/) )
65adantl 277 . . 3  |-  ( ( A  ~<_  (/)  /\  x : A -1-1-> (/) )  ->  A  =  (/) )
71, 6exlimddv 1910 . 2  |-  ( A  ~<_  (/)  ->  A  =  (/) )
8 0ex 4156 . . . 4  |-  (/)  e.  _V
9 domrefg 6812 . . . 4  |-  ( (/)  e.  _V  ->  (/)  ~<_  (/) )
108, 9ax-mp 5 . . 3  |-  (/)  ~<_  (/)
11 breq1 4032 . . 3  |-  ( A  =  (/)  ->  ( A  ~<_  (/) 
<->  (/) 
~<_  (/) ) )
1210, 11mpbiri 168 . 2  |-  ( A  =  (/)  ->  A  ~<_  (/) )
137, 12impbii 126 1  |-  ( A  ~<_  (/) 
<->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2164   _Vcvv 2760   (/)c0 3446   class class class wbr 4029   -->wf 5242   -1-1->wf1 5243    ~<_ cdom 6784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4462
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 5248  df-fn 5249  df-f 5250  df-f1 5251  df-fo 5252  df-f1o 5253  df-en 6786  df-dom 6787
This theorem is referenced by: (None)
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