Theorem ididg 5413

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 2771	. 2 ⊢ 𝐴 = 𝐴
2		ideqg 5411	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 I 𝐴 ↔ 𝐴 = 𝐴))
3	1, 2	mpbiri 248	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 I 𝐴)

Syntax hints:   → wi 4   = wceq 1631   ∈ wcel 2145   class class class wbr 4787   I cid 5157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257

This theorem is referenced by:  issetid  5414  opelresi  5548  fvi  6399  dfpo2  31983