Theorem issetid 4909

Description: Two ways of expressing set existence. (Contributed by NM, 16-Feb-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
issetid	|- ( A e. _V <-> A _I A )

Proof of Theorem issetid

Step	Hyp	Ref	Expression
1		ididg 4908	. 2 |- ( A e. _V -> A _I A )
2		reli 4884	. . 3 |- Rel _I
3	2	brrelex1i 4793	. 2 |- ( A _I A -> A e. _V )
4	1, 3	impbii 126	1 |- ( A e. _V <-> A _I A )

Syntax hints:   <-> wb 105   e. wcel 2203   _Vcvv 2813   class class class wbr 4109   _I cid 4409

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756

This theorem is referenced by: (None)