Theorem bj-elid3 36687

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 3477	. 2 ⊢ 𝑥 ∈ V
2		bj-opelidb1 36673	. 2 ⊢ (⟨𝑥, 𝐴⟩ ∈ I ↔ (𝑥 ∈ V ∧ 𝑥 = 𝐴))
3	1, 2	mpbiran 707	1 ⊢ (⟨𝑥, 𝐴⟩ ∈ I ↔ 𝑥 = 𝐴)

Syntax hints:   ↔ wb 205   = wceq 1533   ∈ wcel 2098  Vcvv 3473  ⟨cop 4638   I cid 5579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-opab 5215  df-id 5580

This theorem is referenced by: (None)