Theorem bj-elid3 37755

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 3457	. 2 ⊢ 𝑥 ∈ V
2		bj-opelidb1 37741	. 2 ⊢ (⟨𝑥, 𝐴⟩ ∈ I ↔ (𝑥 ∈ V ∧ 𝑥 = 𝐴))
3	1, 2	mpbiran 721	1 ⊢ (⟨𝑥, 𝐴⟩ ∈ I ↔ 𝑥 = 𝐴)

Syntax hints:   ↔ wb 209   = wceq 1568   ∈ wcel 2141  Vcvv 3453  ⟨cop 4594   I cid 5555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-opab 5173  df-id 5556

This theorem is referenced by: (None)