Theorem bj-elid3 37839

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 3459	. 2 ⊢ 𝑥 ∈ V
2		bj-opelidb1 37825	. 2 ⊢ (⟨𝑥, 𝐴⟩ ∈ I ↔ (𝑥 ∈ V ∧ 𝑥 = 𝐴))
3	1, 2	mpbiran 721	1 ⊢ (⟨𝑥, 𝐴⟩ ∈ I ↔ 𝑥 = 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ⟨cop 4595   I cid 5555

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-opab 5174  df-id 5556

This theorem is used by: (None)