Theorem issetid 5865

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 5864	. 2 ⊢ (𝐴 ∈ V → 𝐴 I 𝐴)
2		reli 5836	. . 3 ⊢ Rel I
3	2	brrelex1i 5741	. 2 ⊢ (𝐴 I 𝐴 → 𝐴 ∈ V)
4	1, 3	impbii 209	1 ⊢ (𝐴 ∈ V ↔ 𝐴 I 𝐴)

Syntax hints:   ↔ wb 206   ∈ wcel 2108  Vcvv 3480   class class class wbr 5143   I cid 5577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692

This theorem is referenced by: (None)