Theorem issetid 4890

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 4889	. 2 |- ( A e. _V -> A _I A )
2		reli 4865	. . 3 |- Rel _I
3	2	brrelex1i 4775	. 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 2202   _Vcvv 2803   class class class wbr 4093   _I cid 4391

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 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738

This theorem is referenced by: (None)