Theorem bj-elid3 35338

Description: Characterization of the couples in I whose first component is a setvar. (Contributed by BJ, 29-Mar-2020.)

Assertion

Ref	Expression
bj-elid3	⊢ (⟨𝑥, 𝐴⟩ ∈ I ↔ 𝑥 = 𝐴)

Proof of Theorem bj-elid3

Step	Hyp	Ref	Expression
1		vex 3436	. 2 ⊢ 𝑥 ∈ V
2		bj-opelidb1 35324	. 2 ⊢ (⟨𝑥, 𝐴⟩ ∈ I ↔ (𝑥 ∈ V ∧ 𝑥 = 𝐴))
3	1, 2	mpbiran 706	1 ⊢ (⟨𝑥, 𝐴⟩ ∈ I ↔ 𝑥 = 𝐴)

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ⟨cop 4567   I cid 5488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-opab 5137  df-id 5489

This theorem is referenced by: (None)