Theorem issetid 5719

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 5718	. 2 ⊢ (𝐴 ∈ V → 𝐴 I 𝐴)
2		reli 5692	. . 3 ⊢ Rel I
3	2	brrelex1i 5602	. 2 ⊢ (𝐴 I 𝐴 → 𝐴 ∈ V)
4	1, 3	impbii 211	1 ⊢ (𝐴 ∈ V ↔ 𝐴 I 𝐴)

Syntax hints:   ↔ wb 208   ∈ wcel 2110  Vcvv 3494   class class class wbr 5058   I cid 5453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-id 5454  df-xp 5555  df-rel 5556

This theorem is referenced by: (None)