Theorem ididg 5833

Description: A set is identical to itself. (Contributed by NM, 28-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
ididg	⊢ (𝐴 ∈ 𝑉 → 𝐴 I 𝐴)

Proof of Theorem ididg

Step	Hyp	Ref	Expression
1		eqid 2760	. 2 ⊢ 𝐴 = 𝐴
2		ideqg 5831	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 I 𝐴 ↔ 𝐴 = 𝐴))
3	1, 2	mpbiri 261	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 I 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   class class class wbr 5103   I cid 5549

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 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

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 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662

This theorem is used by:  issetid  5834  opelidres  5984  fvi  6954