Theorem ididg 4849

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 2207	. 2 ⊢ 𝐴 = 𝐴
2		ideqg 4847	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 I 𝐴 ↔ 𝐴 = 𝐴))
3	1, 2	mpbiri 168	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 I 𝐴)

Syntax hints:   → wi 4   = wceq 1373   ∈ wcel 2178   class class class wbr 4059   I cid 4353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700

This theorem is referenced by:  issetid  4850  opelresi  4989  fvi  5659