Theorem issetid 4758

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 ∈ V ↔ 𝐴 I 𝐴)

Proof of Theorem issetid

Step	Hyp	Ref	Expression
1		ididg 4757	. 2 ⊢ (𝐴 ∈ V → 𝐴 I 𝐴)
2		reli 4733	. . 3 ⊢ Rel I
3	2	brrelex1i 4647	. 2 ⊢ (𝐴 I 𝐴 → 𝐴 ∈ V)
4	1, 3	impbii 125	1 ⊢ (𝐴 ∈ V ↔ 𝐴 I 𝐴)

Syntax hints:   ↔ wb 104   ∈ wcel 2136  Vcvv 2726   class class class wbr 3982   I cid 4266

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611

This theorem is referenced by: (None)