Theorem risset 2422

Description: Two ways to say "𝐴 belongs to 𝐵." (Contributed by NM, 22-Nov-1994.)

Assertion

Ref	Expression
risset	⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem risset

Step	Hyp	Ref	Expression
1		exancom 1555	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
2		df-rex 2381	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝑥 = 𝐴 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴))
3		df-clel 2096	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
4	1, 2, 3	3bitr4ri 212	1 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴)

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1299  ∃wex 1436   ∈ wcel 1448  ∃wrex 2376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-4 1455  ax-ial 1482

This theorem depends on definitions:  df-bi 116  df-clel 2096  df-rex 2381

This theorem is referenced by:  reueq  2836  reuind  2842  0el  3332  iunid  3815  sucel  4270  reusv3  4319  fvmptt  5444  releldm2  6013  qsid  6424  rerecclap  8351  nndiv  8619  zq  9268  4fvwrd4  9758  bj-bdcel  12616