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Theorem ifr0 36873
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 1863 . . . . 5  |-  x  =  x
2 vex 3034 . . . . . 6  |-  x  e. 
_V
32ideq 4992 . . . . 5  |-  ( x  _I  x  <->  x  =  x )
41, 3mpbir 214 . . . 4  |-  x  _I  x
5 frirr 4816 . . . . 5  |-  ( (  _I  Fr  A  /\  x  e.  A )  ->  -.  x  _I  x
)
65ex 441 . . . 4  |-  (  _I  Fr  A  ->  (
x  e.  A  ->  -.  x  _I  x
) )
74, 6mt2i 122 . . 3  |-  (  _I  Fr  A  ->  -.  x  e.  A )
87eq0rdv 3773 . 2  |-  (  _I  Fr  A  ->  A  =  (/) )
9 fr0 4818 . . 3  |-  _I  Fr  (/)
10 freq2 4810 . . 3  |-  ( A  =  (/)  ->  (  _I  Fr  A  <->  _I  Fr  (/) ) )
119, 10mpbiri 241 . 2  |-  ( A  =  (/)  ->  _I  Fr  A )
128, 11impbii 192 1  |-  (  _I  Fr  A  <->  A  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 189    = wceq 1452    e. wcel 1904   (/)c0 3722   class class class wbr 4395    _I cid 4749    Fr wfr 4795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-id 4754  df-fr 4798  df-xp 4845  df-rel 4846
This theorem is referenced by: (None)
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