Theorem issetid 5842

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 5841	. 2 ⊢ (𝐴 ∈ V → 𝐴 I 𝐴)
2		reli 5815	. . 3 ⊢ Rel I
3	2	brrelex1i 5719	. 2 ⊢ (𝐴 I 𝐴 → 𝐴 ∈ V)
4	1, 3	impbii 212	1 ⊢ (𝐴 ∈ V ↔ 𝐴 I 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2146  Vcvv 3457   class class class wbr 5111   I cid 5557

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670

This theorem is used by: (None)