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Theorem ifr0 27629
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 1688 . . . . 5  |-  x  =  x
2 vex 2959 . . . . . 6  |-  x  e. 
_V
32ideq 5025 . . . . 5  |-  ( x  _I  x  <->  x  =  x )
41, 3mpbir 201 . . . 4  |-  x  _I  x
5 frirr 4559 . . . . 5  |-  ( (  _I  Fr  A  /\  x  e.  A )  ->  -.  x  _I  x
)
65ex 424 . . . 4  |-  (  _I  Fr  A  ->  (
x  e.  A  ->  -.  x  _I  x
) )
74, 6mt2i 112 . . 3  |-  (  _I  Fr  A  ->  -.  x  e.  A )
87eq0rdv 3662 . 2  |-  (  _I  Fr  A  ->  A  =  (/) )
9 fr0 4561 . . 3  |-  _I  Fr  (/)
10 freq2 4553 . . 3  |-  ( A  =  (/)  ->  (  _I  Fr  A  <->  _I  Fr  (/) ) )
119, 10mpbiri 225 . 2  |-  ( A  =  (/)  ->  _I  Fr  A )
128, 11impbii 181 1  |-  (  _I  Fr  A  <->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    = wceq 1652    e. wcel 1725   (/)c0 3628   class class class wbr 4212    _I cid 4493    Fr wfr 4538
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-id 4498  df-fr 4541  df-xp 4884  df-rel 4885
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