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Theorem ifr0 27653
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 1644 . . . . 5  |-  x  =  x
2 vex 2791 . . . . . 6  |-  x  e. 
_V
32ideq 4836 . . . . 5  |-  ( x  _I  x  <->  x  =  x )
41, 3mpbir 200 . . . 4  |-  x  _I  x
5 frirr 4370 . . . . 5  |-  ( (  _I  Fr  A  /\  x  e.  A )  ->  -.  x  _I  x
)
65ex 423 . . . 4  |-  (  _I  Fr  A  ->  (
x  e.  A  ->  -.  x  _I  x
) )
74, 6mt2i 110 . . 3  |-  (  _I  Fr  A  ->  -.  x  e.  A )
87eq0rdv 3489 . 2  |-  (  _I  Fr  A  ->  A  =  (/) )
9 fr0 4372 . . 3  |-  _I  Fr  (/)
10 freq2 4364 . . 3  |-  ( A  =  (/)  ->  (  _I  Fr  A  <->  _I  Fr  (/) ) )
119, 10mpbiri 224 . 2  |-  ( A  =  (/)  ->  _I  Fr  A )
128, 11impbii 180 1  |-  (  _I  Fr  A  <->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    = wceq 1623    e. wcel 1684   (/)c0 3455   class class class wbr 4023    _I cid 4304    Fr wfr 4349
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-fr 4352  df-xp 4695  df-rel 4696
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