Theorem issetid 5752

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 5751	. 2 ⊢ (𝐴 ∈ V → 𝐴 I 𝐴)
2		reli 5725	. . 3 ⊢ Rel I
3	2	brrelex1i 5634	. 2 ⊢ (𝐴 I 𝐴 → 𝐴 ∈ V)
4	1, 3	impbii 208	1 ⊢ (𝐴 ∈ V ↔ 𝐴 I 𝐴)

Syntax hints:   ↔ wb 205   ∈ wcel 2108  Vcvv 3422   class class class wbr 5070   I cid 5479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587

This theorem is referenced by: (None)