Theorem risset 2578

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 1661	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
2		df-rex 2534	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝑥 = 𝐴 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴))
3		df-clel 2234	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
4	1, 2, 3	3bitr4ri 213	1 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴)

This proof depends on syntax axioms:   ∧ wa 104   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∃wrex 2529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-ial 1587

This proof depends on definitions:  df-bi 117  df-clel 2234  df-rex 2534

This theorem is used by:  clel5  2963  reueq  3025  reuind  3031  0el  3544  iunid  4068  sucel  4555  reusv3  4606  fvmptt  5797  releldm2  6419  qsid  6874  rerecclap  9060  nndiv  9345  zq  10026  4fvwrd4  10547  ballotfilemsima  13259  conjnmzb  14083  bj-bdcel  16863