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Theorem ifr0 36803
Description: A class that is founded by the identity relation is null. (Contributed by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ifr0  |-  (  _I  Fr  A  <->  A  =  (/) )

Proof of Theorem ifr0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 equid 1855 . . . . 5  |-  x  =  x
2 vex 3048 . . . . . 6  |-  x  e. 
_V
32ideq 4987 . . . . 5  |-  ( x  _I  x  <->  x  =  x )
41, 3mpbir 213 . . . 4  |-  x  _I  x
5 frirr 4811 . . . . 5  |-  ( (  _I  Fr  A  /\  x  e.  A )  ->  -.  x  _I  x
)
65ex 436 . . . 4  |-  (  _I  Fr  A  ->  (
x  e.  A  ->  -.  x  _I  x
) )
74, 6mt2i 122 . . 3  |-  (  _I  Fr  A  ->  -.  x  e.  A )
87eq0rdv 3769 . 2  |-  (  _I  Fr  A  ->  A  =  (/) )
9 fr0 4813 . . 3  |-  _I  Fr  (/)
10 freq2 4805 . . 3  |-  ( A  =  (/)  ->  (  _I  Fr  A  <->  _I  Fr  (/) ) )
119, 10mpbiri 237 . 2  |-  ( A  =  (/)  ->  _I  Fr  A )
128, 11impbii 191 1  |-  (  _I  Fr  A  <->  A  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 188    = wceq 1444    e. wcel 1887   (/)c0 3731   class class class wbr 4402    _I cid 4744    Fr wfr 4790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-id 4749  df-fr 4793  df-xp 4840  df-rel 4841
This theorem is referenced by: (None)
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