MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eueqi Structured version   Visualization version   GIF version

Theorem eueqi 3686
Description: There exists a unique set equal to a given set. Inference associated with euequ 2684. See euequ 2684 in the case of a setvar. (Contributed by NM, 5-Apr-1995.)
Hypothesis
Ref Expression
eueqi.1 𝐴 ∈ V
Assertion
Ref Expression
eueqi ∃!𝑥 𝑥 = 𝐴
Distinct variable group:   𝑥,𝐴

Proof of Theorem eueqi
StepHypRef Expression
1 eueqi.1 . 2 𝐴 ∈ V
2 eueq 3685 . 2 (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
31, 2mpbi 233 1 ∃!𝑥 𝑥 = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2115  ∃!weu 2654  Vcvv 3480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482
This theorem is referenced by:  eueq2  3687  eueq3  3688  fsn  6886  bj-nuliota  34388  prprval  43897
  Copyright terms: Public domain W3C validator