Theorem issetid 4701

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 4700	. 2 |- ( A e. _V -> A _I A )
2		reli 4676	. . 3 |- Rel _I
3	2	brrelex1i 4590	. 2 |- ( A _I A -> A e. _V )
4	1, 3	impbii 125	1 |- ( A e. _V <-> A _I A )

Syntax hints:   <-> wb 104   e. wcel 1481   _Vcvv 2689   class class class wbr 3937   _I cid 4218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554

This theorem is referenced by: (None)