Theorem bj-elid3 38089

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 3454	. 2 ⊢ 𝑥 ∈ V
2		bj-opelidb1 38075	. 2 ⊢ (⟨𝑥, 𝐴⟩ ∈ I ↔ (𝑥 ∈ V ∧ 𝑥 = 𝐴))
3	1, 2	mpbiran 722	1 ⊢ (⟨𝑥, 𝐴⟩ ∈ I ↔ 𝑥 = 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ⟨cop 4589   I cid 5544

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-opab 5167  df-id 5545

This theorem is used by: (None)