Theorem ideq 2127

Description: For sets, the identity relation is the same as equality.

Hypotheses

Ref	Expression
ideq.1	⊢ A ∈ V
ideq.2	⊢ B ∈ V

Assertion

Ref	Expression
ideq	⊢ (AIB ↔ A = B)

Proof of Theorem ideq

Step	Hyp	Ref	Expression
1		ideq.1	. 2 ⊢ A ∈ V
2		ideq.2	. 2 ⊢ B ∈ V
3		ideqg 2126	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → (AIB ↔ A = B))
4	1, 2, 3	mp2an 520	1 ⊢ (AIB ↔ A = B)

Syntax hints:   ↔ wb 127   = wceq 1091   ∈ wcel 1092  Vcvv 1348   class class class wbr 2054  Icid 2057

This theorem is referenced by:  dmi 2545  iss 2599  imai 2613  intasym 2627  intirr 2628  cnvi 2634  coi1 2665  fcoi1 2765  fcoi2 2766  fvi 2900  ider 3208  idssen 3309

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125

metamath.org